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Chapter 4. Calculating Probabilities: Taking Chances

Life is full of uncertainty.

Sometimes it tin exist impossible to say what volition happen from one infinitesimal to the next. Merely certain events are more likely to occur than others, and that'south where probability theory comes into play. Probability lets you predict the future by assessing how likely outcomes are, and knowing what could happen helps you make informed decisions . In this chapter, you lot'll detect out more about probability and learn how to accept control of the future!

Fatty Dan's K Slam

Fat Dan'south Casino is the well-nigh popular casino in the commune. All sorts of games are offered, from roulette to slot machines, poker to blackjack.

It just and then happens that today is your lucky twenty-four hours. Head Kickoff Labs has given you a whole rack of chips to squander at Fat Dan's, and you get to keep any winnings. Desire to give it a try? Go along—you know you want to.

In that location's a lot of action over at the roulette bike, and some other game is just near to start. Let'southward see how lucky you are.

Scroll up for roulette!

You lot've probably seen people playing roulette in movies even if you've never tried playing yourself. The croupier spins a roulette wheel, so spins a ball in the reverse direction, and you place bets on where yous recollect the ball volition land.

The roulette bike used in Fat Dan's Casino has 38 pockets that the ball can fall into. The main pockets are numbered from ane to 36, and each pocket is colored either reddish or black. There are two extra pockets numbered 0 and 00. These pockets are both green.

You can identify all sorts of bets with roulette. For instance, you tin bet on a particular number, whether that number is odd or even, or the colour of the pocket. You lot'll hear more about other bets when you first playing. One other matter to call up: if the ball lands on a green pocket, y'all lose.

Roulette boards make it easier to keep track of which numbers and colors go together.

Your very own roulette board

You'll be placing a lot of roulette bets in this chapter. Hither's a handy roulette board for y'all to cut out and keep. You lot can utilize it to assistance piece of work out the probabilities in this chapter.

Note

Simply exist careful with those scissors.

Identify your bets now!

Have you cut out your roulette board? The game is just first. Where do you call back the brawl will land? Choose a number on your roulette board, so we'll place a bet.

Right, before placing any bets, information technology makes sense to see how likely information technology is that you'll win.

Maybe some bets are more likely than others. Information technology sounds like we demand to look at some probabilities...

Brain Power

What things practice yous demand to call up about earlier placing whatever roulette bets? Given the choice, what sort of bet would you make? Why?

What are the chances?

Take you ever been in a situation where yous've wondered "Now, what were the chances of that happening?" Perhaps a friend has phoned you at the exact moment you've been thinking almost them, or maybe you've won some sort of raffle or lottery.

Probability is a way of measuring the risk of something happening. You can use it to indicate how likely an occurrence is (the probability that you'll become to sleep some time this week), or how unlikely (the probability that a coyote volition attempt to hit you with an anvil while you're walking through the desert). In stats-speak, an event is any occurrence that has a probability attached to it—in other words, an outcome is any outcome where you can say how likely it is to occur.

Probability is measured on a scale of 0 to ane. If an event is impossible, it has a probability of 0. If it'due south an absolute certainty, and so the probability is i. A lot of the time, you'll be dealing with probabilities somewhere in between.

Hither are some examples on a probability calibration.

Vital Statistics: Effect

An event or occurrence that has a probability assigned to it

Tin you see how probability relates to roulette?

If you know how likely the brawl is to land on a particular number or color, you have some way of judging whether or not you should place a item bet. Information technology'southward useful knowledge if yous want to win at roulette.

Find roulette probabilities

Allow'southward take a closer look at how we calculated that probability.

Here are all the possible outcomes from spinning the roulette cycle. The thing we're really interested in is winning the bet—that is, the ball landing on a 7.

To find the probability of winning, we take the number of ways of winning the bet and divide by the number of possible outcomes similar this:

We can write this in a more general fashion, too. For the probability of any issue A:

South is known every bit the possibility infinite, or sample infinite. It'southward a shorthand way of referring to all of the possible outcomes. Possible events are all subsets of S.

You tin can visualize probabilities with a Venn diagram

Probabilities can chop-chop get complicated, so it'due south frequently very useful to accept some way of visualizing them. 1 way of doing then is to draw a box representing the possibility space Due south , then draw circles for each relevant effect. This sort of diagram is known every bit a Venn diagram. Here'south a Venn diagram for our roulette problem, where A is the event of getting a 7.

Very often, the numbers themselves aren't shown on the Venn diagram. Instead of numbers, yous take the option of using the actual probabilities of each event in the diagram. It all depends on what kind of data yous need to aid yous solve the problem.

Complementary events

There's a shorthand fashion of indicating the event that A does non occur—A^I. A^I is known equally the complementary event of A.

There's a clever way of calculating P(A^I). A^I covers every possibility that'south not in effect A, so betwixt them, A and A^I must cover every eventuality. If something'southward in A, it tin't be in A^I, and if something's not in A, it must be in A^I. This ways that if you add P(A) and P(A^I) together, you lot get 1. In other words, there'south a 100% adventure that something volition be in either A or A^I. This gives u.s.

P(A) + P(A^I) = 1

or

P(A^I) = one – P(A)

It'due south time to play!

A game of roulette is but near to brainstorm.

Look at the events on the previous page. Nosotros'll place a bet on the one that's most likely to occur—that the ball will land in a black pocket.

And the winning number is...

Oh dear! Fifty-fifty though our most likely probability was that the ball would country in a black pocket, it really landed in the green 0 pocket. You lot lose some of your chips.

Probabilities are merely indications of how likely events are; they're not guarantees.

The important thing to remember is that a probability indicates a long-term tendency only. If yous were to play roulette thousands of times, you would expect the ball to land in a blackness pocket in 18/38 spins, approximately 47% of the time, and a light-green pocket in 2/38 spins, or 5% of the time. Even though you'd wait the ball to country in a green pocket relatively infrequently, that doesn't mean information technology tin can't happen.

No matter how unlikely an consequence is, if it's not impossible, it can still happen.

Let's bet on an fifty-fifty more likely event

Let's expect at the probability of an event that should be more likely to happen. Instead of betting that the ball will land in a blackness pocket, let'due south bet that the ball volition land in a blackness or a red pocket. To work out the probability, all we have to do is count how many pockets are blood-red or black, then split by the number of pockets. Sound easy plenty?

We tin use the probabilities we already know to work out the one we don't know.

Take a look at your roulette board. There are only three colors for the ball to land on: reddish, black, or green. As we've already worked out what P(Green) is, we tin can use this value to find our probability without having to count all those black and ruby pockets.

P(Blackness or Ruby)	= P(GreenishI)
	= 1 – P(Light-green)
	= one – 0.053
	= 0.947 (to 3 decimal places)

You lot tin likewise add together probabilities

There'south however some other way of working out this sort of probability. If we know P(Black) and P(Ruby-red), nosotros tin find the probability of getting a blackness or red by adding these two probabilities together. Permit'southward run across.

In this case, calculation the probabilities gives exactly the same consequence equally counting all the red or black pockets and dividing by 38.

Vital Statistics: Probability

To find the probability of an effect A, use

Vital Statistics: A^I

A^I is the complementary outcome of A. It'due south the probability that event A does not occur.

P(A^I) = one – P(A)

You win!

This time the ball landed in a red pocket, the number 7, then you win some fries.

Time for another bet

Now that you're getting the hang of calculating probabilities, let'south attempt something else. What's the probability of the ball landing on a black or even pocket?

Sometimes you can add together probabilities, just it doesn't piece of work in all circumstances.

We might not exist able to count on being able to practise this probability calculation in quite the same manner as the previous one. Try the exercise on the next folio, and come across what happens.

Exclusive events and intersecting events

When we were working out the probability of the brawl landing in a blackness or red pocket, we were dealing with 2 separate events, the ball landing in a black pocket and the ball landing in a cherry pocket. These two events are mutually exclusive considering it's impossible for the ball to land in a pocket that's both black and red.

If two events are mutually exclusive, merely one of the two can occur.

What near the black and even events? This time the events aren't mutually exclusive. It'southward possible that the ball could land in a pocket that's both black and even. The 2 events intersect.

If 2 events intersect, it's possible they can occur simultaneously.

Brain Ability

What sort of event do you think this intersection could accept had on the probability?

Problems at the intersection

Calculating the probability of getting a black or fifty-fifty went wrong because we included blackness and even pockets twice. Here's what happened.

First of all, we found the probability of getting a black pocket and the probability of getting an even number.

When nosotros added the two probabilities together, we counted the probability of getting a black and even pocket twice.

To get the correct reply, nosotros demand to subtract the probability of getting both black and even. This gives us

P(Black or Even) = P(Blackness) + P(Even) – P(Blackness and Fifty-fifty)

Nosotros can now substitute in the values we just calculated to find P(Blackness or Even):

P(Black or Even) = 18/38 + 18/38 – 10/38 = 26/38 = 0.684

Some more notation

There's a more full general fashion of writing this using some more math autograph.

Offset of all, we can apply the notation A ∩ B to refer to the intersection between A and B. Yous can call up of this symbol as meaning "and." It takes the common elements of events.

A ∪ B, on the other hand, means the union of A and B. It includes all of the elements in A and also those in B. You can recollect of it as meaning "or."

If A ∪ B =1, then A and B are said to be exhaustive. Between them, they make upwardly the whole of Due south. They frazzle all possibilities.

Information technology's not actually that different.

Mutually exclusive events accept no elements in common with each other. If you accept two mutually exclusive events, the probability of getting A and B is actually 0—so P(A ∩ B) = 0. Permit's revisit our black-or-ruddy example. In this bet, getting a cerise pocket on the roulette wheel and getting a black pocket are mutually exclusive events, as a pocket tin can't be both red and black. This means that P(Black ∩ Red) = 0, and then that office of the equation just disappears.

Watch it!

At that place's a difference between sectional and exhaustive.

If events A and B are exclusive, so

P(A ∩ B) = 0

If events A and B are exhaustive, then

P(A ∪ B) = one

Vital Statistics: A or B

To observe the probability of getting event A or B, apply

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

∪ means OR

∩ means AND

Another unlucky spin...

We know that the probability of the ball landing on black or even is 0.684, only, unfortunately, the brawl landed on 23, which is red and odd.

...merely information technology's fourth dimension for another bet

Even with the odds in our favor, nosotros've been unlucky with the outcomes at the roulette table. The croupier decides to take pity on united states and offers a little within data. After she spins the roulette wheel, she'll requite u.s.a. a inkling well-nigh where the ball landed, and we'll work out the probability based on what she tells the states.

Should nosotros take this bet?

How does the probability of getting even given that nosotros know the brawl landed in a black pocket compare to our concluding bet that the brawl would land on blackness or fifty-fifty. Let'south effigy information technology out.

Conditions apply

The croupier says the ball has landed in a black pocket. What's the probability that the pocket is also even?

This is a slightly different trouble

We don't want to find the probability of getting a pocket that is both blackness and even, out of all possible pockets. Instead, we desire to notice the probability that the pocket is even, given that we already know it's black.

In other words, we want to discover out how many pockets are even out of all the black ones. Out of the xviii blackness pockets, 10 of them are even, so

It turns out that even with some within data, our odds are really lower than before. The probability of even given black is really less than the probability of black or even.

However, a probability of 0.556 is nevertheless better than fifty% odds, so this is still a pretty expert bet. Let's go for it.

Find conditional probabilities

So how tin we generalize this sort of problem? Starting time of all, nosotros need some more notation to represent conditional probabilities, which measure the probability of one upshot occurring relative to some other occurring.

If we want to express the probability of 1 effect happening given another 1 has already happened, nosotros use the "|" symbol to hateful "given." Instead of saying "the probability of result A occurring given event B," we can shorten information technology to say

P(A | B)

Annotation

The probability of A give that we know B has happened.

So at present we need a general way of calculating P(A | B). What we're interested in is the number of outcomes where both A and B occur, divided by all the B outcomes. Looking at the Venn diagram, we get:

We can rewrite this equation to give u.s. a mode of finding P(A ∩ B)

P(A ∩ B) = P(A | B) × P(B)

It doesn't cease there. Another way of writing P(A ∩ B) is P(B ∩ A). This means that we tin rewrite the formula as

P(B ∩ A) = P(B | A) × P(A)

In other words, just flip around the A and the B.

Venn diagrams aren't e'er the best way of visualizing conditional probability.

Don't worry, there's another sort of diagram you can use—a probability tree.

You can visualize conditional probabilities with a probability tree

It's not ever easy to visualize conditional probabilities with Venn diagrams, but there'due south another sort of diagram that actually comes in handy in this situation—the probability tree. Here's a probability tree for our trouble with the roulette wheel, showing the probabilities for getting different colored and odd or fifty-fifty pockets.

The starting time set of branches shows the probability of each result, and so the probability of getting a black is xviii/38, or 0.474. The second set of branches shows the probability of outcomes given the outcome of the branch it is linked to . The probability of getting an odd pocket given we know information technology's blackness is viii/eighteen, or 0.444.

Trees also help you summate conditional probabilities

Probability trees don't simply assist you visualize probabilities; they can help yous to summate them, likewise.

Let's take a general look at how you tin can do this. Here'due south another probability tree, this time with a dissimilar number of branches. It shows two levels of events: A and A^I and B and B^I. A^I refers to every possibility non covered past A, and B^I refers to every possibility non covered by B.

You can find probabilities involving intersections past multiplying the probabilities of linked branches together. Equally an case, suppose y'all want to detect P(A ∩ B). You can detect this by multiplying P(B) and P(A | B) together. In other words, you multiply the probability on the first level B co-operative with the probability on the second level A branch.

Using probability trees gives you the same results you saw earlier, and information technology's up to you whether you use them or not. Probability trees can be time-consuming to draw, but they offer you a way of visualizing conditional probabilities.

Vital Statistics: Conditions

Bad luck!

You lot placed a bet that the brawl would land in an fifty-fifty pocket given we've been told information technology's blackness. Unfortunately, the brawl landed in pocket 17, so you lose a few more than chips.

Mayhap we tin can win some chips back with another bet. This time, the croupier says that the brawl has landed in an even pocket. What'southward the probability that the pocket is also black?

Note

This is the opposite of the previous bet.

We can reuse the probability calculations we already did.

Our previous task was to effigy out P(Even | Black), and we can utilise the probabilities we found solving that problem to calculate P(Black | Even). Hither's the probability tree nosotros used before:

We can discover P(Blackness l Even) using the probabilities nosotros already take

And then how do we find P(Black | Even)? In that location'southward notwithstanding a style of computing this using the probabilities we already have even if it's not immediately obvious from the probability tree. All we have to practice is look at the probabilities we already take, and use these to somehow calculate the probabilities we don't yet know.

Permit'south start off by looking at the overall probability nosotros need to find, P(Black | Even).

Using the formula for finding provisional probabilities, we have

If nosotros can detect what the probabilities of P(Blackness ∩ Even) and P(Even) are, nosotros'll be able to use these in the formula to calculate P(Black | Even). All we need is some mechanism for finding these probabilities.

Sound hard? Don't worry, we'll guide you through how to practise information technology.

Use the probabilities you take to calculate the probabilities you need

Stride 1: Finding P(Blackness ∩ Fifty-fifty)

Permit's start off with the outset part of the formula, P(Blackness ∩ Fifty-fifty).

So where does this get usa?

Nosotros want to find the probability P(Black | Even). We tin can exercise this by evaluating

Brain Power

Take some other await at the probability tree in So where does this get united states of america?. How practise you think we can use it to notice P(Even)?

Step 2: Finding P(Even)

The side by side step is to detect the probability of the ball landing in an fifty-fifty pocket, P(Even). We can discover this by considering all the ways in which this could happen.

A ball can land in an fifty-fifty pocket by landing in either a pocket that's both black and fifty-fifty, or in a pocket that's both red and fifty-fifty. These are all the possible ways in which a ball can land in an even pocket.

This means that we notice P(Even) past calculation together P(Black ∩ Even) and P(Red ∩ Even). In other words, we add the probability of the pocket being both black and even to the probability of it beingness both ruby-red and even. The relevant branches are highlighted on the probability tree.

Step three: Finding P(Black l Fifty-fifty)

Can you remember our original problem? We wanted to observe P(Black | Even) where

Putting these together means that we can calculate P(Blackness | Even) using probabilities from the probability tree

This means that nosotros at present have a manner of finding new provisional probabilities using probabilities we already know—something that can assist with more complicated probability problems.

Let'due south look at how this works in general.

These results tin can be generalized to other bug

Imagine yous have a probability tree showing events A and B like this, and assume you lot know the probability on each of the branches.

Now imagine you want to discover P(A | B), and the information shown on the branches above is all the information that you take. How can y'all use the probabilities you have to work out P(A | B)?

We tin can start with the formula we had before:

Now nosotros can find P(A ∩ B) using the probabilities we take on the probability tree. In other words, we can calculate P(A ∩ B) using

P(A ∩ B) = P(A) × P(B | A)

Simply how practise nosotros find P(B)?

Brain Ability

Take a good look at the probability tree. How would you apply it to observe P(B)?

Employ the Law of Full Probability to find P(B)

To find P(B), we utilise the aforementioned procedure that nosotros used to find P(Even) earlier; we need to add together the probabilities of all the different ways in which the event we want tin can possibly happen.

There are two ways in which even B can occur: either with event A, or without it. This means that we can find P(B) using:

P(B) = P(A ∩ B) + P(A^I ∩ B)

Notation

Add together both of the intersections to get P(B).

We can rewrite this in terms of the probabilities we already know from the probability tree. This means that we tin can employ:

P(A ∩ B) = P(A) × P(B | A)

P(A^I ∩ B) = P(A^I) × P(B | A^I)

This gives us:

P(B) = P(A) × P(B | A) + P(A^I) × P(B | A^I)

This is sometimes known as the Police force of Total Probability, as it gives a way of finding the total probability of a item upshot based on provisional probabilities.

Now that we accept expressions for P(A ∩ B) and P(B), nosotros can put them together to come up up with an expression for P(A | B).

Introducing Bayes' Theorem

Nosotros started off past wanting to find P(A | B) based on probabilities we already know from a probability tree. We already know P(A), and we likewise know P(B | A) and P(B | A^I). What we need is a full general expression for finding conditional probabilities that are the opposite of what nosotros already know, in other words P(A | B).

We started off with:

Relax

Bayes' Theorem is one of the most difficult aspects of probability.

Don't worry if information technology looks complicated—this is as tough as it's going to become. And even though the formula is tricky, visualizing the problem can help.

This is called Bayes' Theorem. It gives you a means of finding reverse provisional probabilities, which is actually useful if you don't know every probability up front.

Vital Statistics: Law of Full Probability

If you have two events A and B, then

P(B)	= P(B ∩ A) + P(B ∩ AI)
	= P(A) P(B | A) + P(AI) P(B | AI)

The Constabulary of Total Probability is the denominator of Bayes' Theorem.

Vital Statistics: Bayes' Theorem

If you accept due north mutually exclusive and exhaustive events, A¹ through to Aⁿ, and B is another event, so

We have a winner!

Congratulations, this fourth dimension the ball landed on ten, a pocket that's both black and even. Y'all've won back some chips.

Information technology's time for one last bet

Before yous leave the roulette tabular array, the croupier has offered you a groovy deal for your concluding bet, triple or naught. If you bet that the ball lands in a black pocket twice in a row, you could win back all of your chips.

Here's the probability tree. Notice that the probabilities for landing on ii blackness pockets in a row are a bit unlike than they were in our probability tree in Bad luck!, where we were trying to summate the likelihood of getting an even pocket given that nosotros knew the pocket was black.

If events affect each other, they are dependent

The probability of getting blackness followed by blackness is a slightly different problem from the probability of getting an even pocket given nosotros already know it'southward black. Take a expect at the equation for this probability:

P(Fifty-fifty | Black) = 10/18 = 0.556

For P(Fifty-fifty | Black), the probability of getting an even pocket is affected by the issue of getting a blackness. We already know that the ball has landed in a black pocket, so we use this knowledge to work out the probability. Nosotros look at how many of the pockets are even out of all the black pockets.

If we didn't know that the ball had landed on a black pocket, the probability would be dissimilar. To work out P(Fifty-fifty), we look at how many pockets are even out of all the pockets

P(Even) = xviii/38 = 0.474

Annotation

These 2 probabilities are dissimilar

P(Fifty-fifty | Black) gives a unlike result from P(Even). In other words, the knowledge nosotros have that the pocket is black changes the probability. These ii events are said to be dependent.

In general terms, events A and B are said to exist dependent if P(A | B) is different from P(A). It'south a way of maxim that the probabilities of A and B are affected by each other.

Brain Power

Await at the probability tree on the previous page again. What do you notice about the sets of branches? Are the events for getting a black in the beginning game and getting a blackness in the second game dependent? Why?

If events do not touch on each other, they are independent

Not all events are dependent. Sometimes events remain completely unaffected by each other, and the probability of an event occurring remains the same irrespective of whether the other event happens or not. Every bit an example, take a expect at the probabilities of P(Black) and P(Black | Black). What practise you notice?

P(Black) = xviii/38 = 0.474

P(Black | Black) = xviii/38 = 0.474

Note

These probabilities are the same. The events are independent.

These ii probabilities have the aforementioned value. In other words, the event of getting a black pocket in this game has no bearing on the probability of getting a black pocket in the next game. These events are independent.

Independent events aren't affected by each other. They don't influence each other'south probabilities in any manner at all. If one event occurs, the probability of the other occurring remains exactly the same.

If events A and B are independent, then the probability of event A is unaffected by event B. In other words

P(A | B) = P(A)

for independent events.

We tin can also utilize this as a test for independence. If you have two events A and B where P(A | B) = P(A), and then the events A and B must exist contained.

More on calculating probability for independent events

Information technology'southward easier to work out other probabilities for contained events too, for example P(A ∩ B).

We already know that

If A and B are independent, P(A | B) is the same equally P(A). This means that

or

P(A ∩ B) = P(A) × P(B)

Sentry it!

If A and B are mutually exclusive, they can't exist independent, and if A and B are contained, they can't be mutually sectional.

If A and B are mutually exclusive, then if event A occurs, result B cannot. This means that the upshot of A affects the outcome of B, and so they're dependent.

Similarly if A and B are independent, they tin't be mutually sectional.

for contained events. In other words, if 2 events are independent, then you can piece of work out the probability of getting both events A and B by multiplying their private probabilities together.

Vital Statistics: Independence

If two events A and B are independent, then

P(A | B) = P(A)

If this holds for any two events, then the events must be independent. Also

P(A ∩ B) = P(A) x P(B)

Winner! Winner!

On both spins of the wheel, the ball landed on xxx, a red square, and you doubled your winnings.

You've learned a lot almost probability over at Fatty Dan's roulette table, and yous'll notice this knowledge will come in handy for what'south alee at the casino. It's a pity you didn't win enough chips to accept whatever home with you lot, though.

Note

[Note from Fat Dan: That'south a relief.]

Besides the chances of winning, you besides demand to know how much yous stand to win in order to decide if the bet is worth the risk.

Betting on an consequence that has a very low probability may be worth information technology if the payoff is loftier enough to compensate y'all for the run a risk. In the next chapter, we'll look at how to gene these payoffs into our probability calculations to help us make more informed betting decisions.

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Source: https://www.oreilly.com/library/view/head-first-statistics/9780596527587/ch04.html

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