Griffiths D. Head First Statistics

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calculating probabilities

The probability of getting black followed by black is a slightly

different problem from the probability of getting an even

pocket given we already know it’s black. Take a look at the

equation for this probability:

P(Even | Black) = 10/18 = 0.556

For P(Even | Black), the probability of getting an even pocket

is affected by the event of getting a black. We already know

that the ball has landed in a black pocket, so we use this

knowledge to work out the probability. We 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 different. To work out P(Even), we

look at how many pockets are even out of all the pockets

P(Even) = 18/38 = 0.474

P(Even | Black) gives a different result from P(Even). In other

words, the knowledge we have that the pocket is black changes

the probability. These two events are said to be dependent.

In general terms, events A and B are said to be dependent if

P(A | B) is different from P(A). It’s a way of saying that the

probabilities of A and B are affected by each other.

These two probabilities

are different

You being here

changes everything.

I’m different when

I’m with you.

Look 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

first game and getting a black in the second game

dependent? Why?

If events affect each other, they are dependent

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Well, you make no

difference to me either. I

don’t care whether you’re

there or not. I guess this

means we’re independent

You think I care about

your outcomes? They’re

irrelevant to me. I just carry

on like you’re not there.

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. As an example, take a look at the probabilities of P(Black) and

P(Black | Black). What do you notice?

P(Black) = 18/38 = 0.474

P(Black | Black) = 18/38 = 0.474

These two probabilities have the same 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’s probabilities in any way at all. If one event occurs, the

probability of the other occurring remains exactly the same.

P(A | B) = P(A)

If events A and B are independent, then the probability of event A is

unaffected by event B. In other words

These probabilities are the same.

The events are independent.

for independent events.

We can also use this as a test for independence. If you have two events

A and B where P(A | B) = P(A), then the events A and B must be

independent.

independent events

If events do not affect each other, they are independent

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calculating probabilities

for independent events. In other words, if two events are

independent, then you can work out the probability of

getting both events A and B by multiplying their individual

probabilities together.

P(A ∩ B) = P(A)

P(B)

It’s easier to work out other probabilities for independent

events too, for example P(A ∩ B).

We already know that

P(A | B) = P(A ∩ B)

P(B)

If A and B are independent, P(A | B) is the same as P(A).

This means that

P(A) = P(A ∩ B)

P(B)

It’s time to calculate another probability. What’s the probability of

the ball landing in a black pocket twice in a row?

More on calculating probability for independent events

If A and B are

mutually exclusive,

they can’t be

independent, and

if A and B are

independent, they can’t be

mutually exclusive.

If A and B are mutually exclusive,

then if event A occurs, event

B cannot. This means that the

outcome of A affects the outcome of

B, and so they’re dependent.

Similarly if A and B are independent,

they can’t be mutually exclusive.

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What’s the difference between

being independent and being mutually

exclusive?

A: Imagine you have two events, A and B.

If A and B are mutually exclusive, then if

event A happens, B cannot. Also, if event B

happens, then A cannot. In other words, it’s

impossible for both events to occur.

If A and B are independent, then the

outcome of A has no effect on the outcome

of B, and the outcome of B has no effect on

the outcome of A. Their respective outcomes

have no effect on each other.

Do both events have to be

independent? Can one event be

independent and the other dependent?

A: No. The two events are independent

of each other, so you can’t have two events

where one is dependent and the other one is

independent.

Are all games on a roulette wheel

independent? Why?

A: Yes, they are. Separate spins of the

roulette wheel do not influence each other.

In each game, the probabilities of the ball

landing on a red, black, or green remain the

same.

You’ve shown how a probability

tree can demonstrate independent events.

How do I use a Venn diagram to tell if

events are independent?

A: A Venn diagram really isn’t the

best way of showing dependence. Venn

diagrams are great if you need to examine

intersections and show mutually exclusive

events. They’re not great for showing

independence though.

It’s time to calculate another probability. What’s the probability of

the ball landing in a black pocket twice in a row?

We need to find P(Black in game 1 ∩ Black in game 2). As the events are independent, the result is

18/38 x 18/38 = 324/1444

= 0.224 (to 3 decimal places)

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)

Vital StatisticsVital Statistics

sharpen solution

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calculating probabilities

The Case of the Two Classes

The Head First Health Club prides itself on its ability to find a class for

everyone. As a result, it is extremely popular with both young and old.

The Health Club is wondering how best to market its new yoga class,

and the Head of Marketing wonders if someone who goes swimming

is more likely to go to a yoga class. “Maybe we could offer some sort of

discount to the swimmers to get them to try out yoga.”

The CEO disagrees. “I think you’re wrong,” he says. “I think

that people who go swimming and people who go to yoga are

independent. I don’t think people who go swimming are any

more likely to do yoga than anyone else.”

They ask a group of 96 people whether they go to the swimming

or yoga classes. Out of these 96 people, 32 go to yoga and 72 go

swimming. 24 people are exceptionally eager and go to both.

So who’s right? Are the yoga and swimming classes

dependent or independent?

Five Minute

Mystery

Five Minute

Mystery

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Tonight’s talk: Dependent and Independent discuss their

differences

Dependent:

Independent, glad you could show up. I’ve been

wanting to catch up with you for some time.

Well, I hear you keep getting fledgling statisticians

into trouble. They’re doing fine until you show up,

and then, whoa, wrong probabilities all over the

place! That ∩ guy has a particularly poor opinion

of you.

It’s that simplistic attitude of yours that gets people

into trouble. They think, “Hey, that Independent

guy looks easy. I’ll just use him for this probability.”

The next thing you know, ∩ has his probabilities all

in a twist. That’s just not the right way of dealing

with dependent events.

You don’t understand the seriousness of the

situation. If people use your way of calculating ∩’s

probability, and the events are dependent, they’re

guaranteed to get the wrong answer. That’s just not

good enough. For dependent events, you only

get the right answer if you take that | guy into

account—he’s a given.

Independent:

Really, Dependent? How come?

I’m a little hurt that ∩’s been saying bad things

about me; I thought I made life easy for him.

You want to work out the probability of getting

two independent events? Easy! Just multiply the

probabilities for the two events together and job

done.

You’re blowing this all out of proportion. Even if

people do decide to use me instead of you, I don’t

see that it can make all that much difference.

I can’t say I pay all that much attention to him.

With independent events, probabilities just turn out

the same.

fireside chat: dependent versus independent

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calculating probabilities

Dependent:

You’re doing it again; you’re oversimplifying things.

Well, I’ve had enough. I think that people need to

think of me first instead of you; that would sort out

all of these problems.

By really thinking through whether events are

dependent or not. Let me give you an example.

Suppose you have a deck of 52 cards, and thirteen

of them are diamonds. Imagine you choose a card

at random and it’s a diamond. What would be the

probability of that happening?

What if you pick out a second card? What’s the

probability of pulling out a second diamond?

No! The events are dependent. You can no longer

say there are 13 diamonds in a pack of 52 cards.

You’ve just removed one diamond, so there are

12 diamonds left out of 51 cards. The probability

drops to 12/51, or 4/17.

But they weren’t. When people think about you

first, it leads them towards making all sorts of

inappropriate assumptions. No wonder ∩ gets so

messed up.

Think nothing of it. Just make sure you think things

through a bit more carefully next time.

Independent:

Yeah? Like how?

That’s easy. It’s 13/52, or 1/4.

It’s the same isn’t it? 1/4.

Not fair, I assumed you put the first card back!

That would have meant the probability of getting a

diamond would have been the same as before, and I

would have been right. The events would have been

independent.

Well, thanks for the chat, Dependent, I’m glad we

had a chance to sort things out.

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Solved: The Case of the Two Classes

Are the yoga and swimming classes dependent or

independent?

The CEO’s right—the classes are independent.

Here’s how he knows.

32 people out of 96 go to yoga classes, so

P(Yoga) = 1/3

72 people go swimming, so

P(Swimming) = 3/4

24 people go to both classes, so

P(Yoga ∩ Swimming) = 1/4

So how do we know the classes are independent? Let’s multiply

together P(Yoga) and P(Swimming) and see what we get.

P(Yoga) × P(Swimming) = 1/3

3/4

= 1/4

As this is the same as P(Yoga ∩ Swimming), we know that the

classes are independent.

Five Minute

Mystery

Solved

Five Minute

Mystery

Solved

five minute mystery solution

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calculating probabilities

Here are a bunch of situations and events. Your task is to say which of

these are dependent, and which are independent.

Dependent

Independent

Throwing a coin and getting heads twice

in a row.

Removing socks from a drawer until you

find a matching pair.

Choosing chocolates at random from a box

and picking dark chocolates twice in a row.

Choosing a card from a deck of cards, and

then choosing another one.

Choosing a card from a deck of cards,

putting the card back in the deck, and then

choosing another one.

The event of getting rain given it’s a

Thursday.

Dependent or

Independent?

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Here are a bunch of situations and events. Your task was to say which

of these are dependent, and which are independent.

Dependent

Independent

Throwing a coin and getting heads twice

in a row.

Removing socks from a drawer until you

find a matching pair.

Choosing chocolates at random from a box

and picking dark chocolates twice in a row.

Choosing a card from a deck of cards, and

then choosing another one.

Choosing a card from a deck of cards,

putting the card back in the deck, and then

choosing another one.

The event of getting rain given it’s a

Thursday.

Dependent or

Independent?

Solution

The second coin throw isn’t

affected by the first.

When you remove one sock, there are fewer socks to choose

from the next time, and this affects the probability.

It’s no more or less likely to rain just

because it’s Thursday, so these two events

are independent.

dependent or independent solution