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Solution: B

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Chapter 15Probability Rules!

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The General Addition Rule

When two events A and B are disjoint, we can use the addition rule for disjoint events from Chapter 14:

P(A B) = P(A) + P(B) However, when our events are not disjoint, this

earlier addition rule will double count the probability of both A and B occurring. Thus, we need the General Addition Rule.

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The General Addition Rule (cont.)

General Addition Rule: For any two events A and B,

P(A B) = P(A) + P(B) – P(A B) The following Venn diagram shows a situation in

which we would use the general addition rule:

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Example: A survey of college students found that 56% live in a campus residence hall, 62% participate in a campus meal program and 42% do both. What is the probability that a randomly selected student either lives or eats on campus?

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Example: Draw a Venn Diagram for the previous example. What is the probability that a randomly selected student:

a. Lives off campus and doesn’t have a meal program?

b. Lives in a residence hall but doesn’t have a meal program?

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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 has already occurred.”

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

P(B|A)P(A B)P(A)

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The General Multiplication Rule

When two events A and B are independent, we can use the multiplication rule for independent events from Chapter 14:

P(A B) = P(A) x P(B) However, when our events are not independent,

this earlier multiplication rule does not work. Thus, we need the General Multiplication Rule.

P(A B) = P(A) P(B|A) or

P(A B) = P(B) P(A|B)

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Example: While dining in a campus facility open only to students with meal plans, you meet someone interesting What is the probability that your new acquaintance lives on campus?

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Independence

Independence of two events means that the outcome of one event does not influence the probability of the other. Events A and B are independent whenever P(B|A) = P(B). (Equivalently, events A and B are independent whenever P(A|B) = P(A).)

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Independent ≠ Disjoint

Give an example of disjoint events: Given an example of independent events: Disjoint events cannot be 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.

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Example: Are living on campus and have a meal plan independent? Are they disjoint?

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Depending on Independence

It’s much easier to think about independent events than to deal with conditional probabilities. It seems that most people’s natural intuition for

probabilities breaks down when it comes to conditional probabilities.

Don’t fall into this trap: whenever you see probabilities multiplied together, stop and ask whether you think they are really independent.

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Example: It is reported that typically as few as 10% of random phone calls result in a completed interview. Reasons are varied, but some of the most common include no answer, refusal to cooperate and failure to complete the call. Which of the following events are independent, disjoint or neither independent nor disjoint?

a. A = Your telephone number is randomly selected.

B = You’re not at home at dinner time when they call.

b. A = As a selected subject, you complete the interview.

B = As a selected subject, you refuse to cooperate.

c. A = You are not at home when they call at 11 am.

B = You are employed full time.

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Drawing Without Replacement

Sampling without replacement means that once one individual is drawn it doesn’t go back into the pool. We often sample without replacement, which

doesn’t matter too much when we are dealing with a large population.

However, when drawing from a small population, we need to take note and adjust probabilities accordingly.

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Example: You draw two cards from a deck and place them in your hand. What is the probability both cards are black face cards?

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Tree Diagrams

A tree diagram helps us think through conditional probabilities by showing sequences of events as paths that look like branches of a tree.

Making a tree diagram for situations with conditional probabilities is consistent with our “make a picture” mantra.

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Tree Diagrams (cont.)

Figure 15.5 is an example of a tree diagram and shows how we multiply the probabilities of the branches together.

All the final outcomes are disjoint and must add up to one.

We can add the final probabilities to find probabilities of compound events.

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Reversing the Conditioning

Reversing the conditioning of two events is rarely intuitive. Suppose we want to know P(A|B), and we know only P(A), P(B), and P(B|A).

We also know P(A B), since

P(A B) = P(A) x P(B|A) From this information, we can find P(A|B):

P(A|B)P(A B)P(B)

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Bayes’s Rule

When we reverse the probability from the conditional probability that you’re originally given, you are actually using Bayes’s Rule.

P B | A P A | B P B P A | B P B P A | BC P BC

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Homework: Pg. 361 1-19 odd (Day 1)

Pg. 361 21-45 odd (Day 2)