Applying the ideas: Probability

Vocabulary

probability

trial

outcome

event

The long-run proportion of an event’s occurrence.

A single attempt of a random phenomenon.

What you measure in a trial.

A collection of outcomes.

More vocabulary

independent

Law of Large Numbers

complement

disjoint

Two events are independent if they don’t affect each other.

In the long run, the frequency of repeated trials approaches the true probability.

The event that does NOT happen

Nothing in common.

Notation alert – Conditional probability

Earlier this year, we looked at contingency tables and talked about conditional distributions.

When we want the probability of an event from a conditional distribution, we write P(B|A) and pronounce it “the probability of B given A.”

A probability that takes into account a given condition is called a conditional probability.

Here are the same classes of students, but now use probability notation:

If I pick a student at random,

What is P(male)?

What is P(sports | male)?

What is P(female | no sports)

Male Female

Sports 25 15 40

No

Sports21 20 41

46 35 81

46

81

25

46

20

41

Disjoint events

Events that do not share outcomes are disjoint.Some examples:

Die roll: “rolling a 6” and “rolling an odd number.”

Card draw: “drawing a Heart” and “drawing a black card”

For any disjoint events, P(A and B) = 0

Independent events

Two events are independent if the outcome of one has no influence on the other.

In formal notation: P(A)= P(A|B) The probability of A is the same as

the probability of A, given that B has already happened.

IT DOESN’T MATTER THAT B HAS ALREADY HAPPENED!

Independent ≠ Disjoint

Disjoint events cannot be independent! Well, why not? Since we know that disjoint events have no outcomes

in common, knowing that one occurred means the other didn’t.

Thus, the probability of the second occurring changed based on our knowledge that the first occurred.

It follows, then, that the two events are not independent.

A common error is to treat disjoint events as if they were independent, and apply the Multiplication Rule for independent events—don’t make that mistake.

Here is a table of teachers:

Are the events “picking a Democrat” and “picking a right hander” disjoint?

Is P(Dem. and righty) = 0?

Are the events “picking a Democrat” and “picking a right hander” independent?

Is P(Dem.) = P(Dem.|righty)?

Left-handed

Right-handed

Democrat 7 15 22

Republican 2 10 12

Other 3 16 19

12 41 53

Key rules – notation alert!

Not (complement)

P(not A) = P(Ac) = 1 – P(A) And (intersection)

P(A and B) = P(A ∩ B) = P(A)P(B|A) Or (union)

P(A or B) = P(A U B) = P(A)+P(B)– P(A and B)

Practice 1

You roll a fair die three times. What is the probability…

a) You roll all 6’s?

b) You roll all odd numbers?

c) None of your rolls gets a multiple of 3?

d) You roll at least one 5?

e) The numbers you roll are NOT all 5’s?

Practice 2

For each of the following, list the sample space and tell whether you think the events are equally likely:

a) Roll two dice; record the sum of the numbersb) A family has three kids; record their genders in

order of birthc) Toss four coins; record the number of tailsd) Toss a coin 10 times; record the longest run of

heads

Practice 3: Using Venn diagrams

Suppose the probability that a US resident has traveled to Canada is 0.18, to Mexico 0.09, and to both 0.04. What is the probability that an American chosen at random has…

a) Traveled to Canada but not Mexico?b) Traveled to either Canada or Mexico?c) Not traveled to either country?

Practice 4

Employment data at a large company reveal that 72% of the workers are married, 44% are college graduates, and half the college grads are married. Find the probability that a randomly chosen employee is

a) Neither married nor a college grad

b) Married but not a college grad

c) Married and a college grad

Practice 5

The local animal shelter reports that it has 24 dogs and 18 cats available for adoption. Eight of the dogs and six of the cats are male. Find these probabilities:

a) The pet is male, given that it is a cat.b) The pet is a cat, given that it is female.c) The pet is female, given that it is a dog.

Choosing without replacement: events that are not independent

What is the probability of drawing two red cards out of a standard deck without replacing the cards after each draw?

What is the probability of drawing a pair of green socks (again without replacement) out of a drawer that contains 7 green and 5 red socks?

26 25.245

52 51

7 6.318

12 11

Using tree diagrams

These are useful when there are multiple events involving conditional probabilities.

They are the most likely type of probability question to appear on an AP test.

So…

Practice 6Suppose that a polygraph (lie detector) can

correctly detect 70% of all lies, but also incorrectly identifies 20% of true statements as lies.

And let’s say that 98% of all job applicants are trustworthy.

They ask all applicants “Have you ever stolen anything from your place of work?” [all applicants would of course answer “no.”]

Given that a job applicant was rejected, what is the probability that she is in fact trustworthy?

Practice 7

Leah is flying from Boston to Denver with a connection in Chicago.

The probability her first flight leaves on time is 0.15.

If that flight is on time, the probability that her luggage will make the connecting flight in Chicago is 0.95.

If the first flight is delayed, the probability that the luggage makes is only 0.65.

Practice 7 (cont.)

There are two events: first flight’s departure and luggage connection. Are they independent?

What is the probability that her luggage arrives with her in Denver?

If Leah arrives in Denver without her luggage, what is probability that her first flight was delayed?