Chapter 15: Probability Rules! AP Statistics

Vocab (quick) Probability: Long-run relative frequency Event: any set or collection of outcomes Sample Space: The collection of all possible outcomes

Vocab (quick) Independent Events: Two events are independent if knowing whether one event occurs does not alter the probability that the other event occurs. Disjoint Events (Mutually Exclusive): Two events are disjoint if they share no outcomes in common. If A and B are disjoint, then knowing A occurs tells us that B cannot occur.

Independence ≠ Disjoint The events are disjoint or mutually exclusive. They have no outcomes in common. Therefore, the outcome of one must change the probability of the other occurring.

Independence ≠ Disjoint Mutually exclusive events can’t be independent. They have no outcomes in common, so knowing that one occurred means that other didn’t. A common error is to treat disjoint events as if they were independent and apply the multiplication rule for independent events.

Law of Large Numbers The Law of Large Numbers states that the long-run relative frequency of repeated independent events settles down to the true relative frequency as the number of trials increases.

Three General Formal Probability Rules A probability is a number between 0 and 1 The probability of the set of all possible outcomes must be 1. The probability of an event occurring is 1 minus the probability that it doesn’t occur.

General Addition Rule In Chapter 14, we had the addition rule (or) for disjoint events We can, however, expand it to any non-disjoint situation.

General Addition Rule The General Addition Rule does not require us to have disjoint sets when we find the probability of A or B.

General Addition Rule

General Addition Rule In a certain area of a city, 45% of residents have brown hair, 58% have brown eyes and 30% have both. What is the probability that a person randomly selected from that area will have brown hair or brown eyes? (PICTURE ALSO)

The General Multiplication Rule In order to use the Multiplication Rule from Chapter 14, events needed to be independent. (Independence Assumption)

The General Multiplication Rule The General Multiplication Rule does not require us to have independent events when we find the probability of A and B. “The probability of A and B is equal to the probability of A times the probability of B given A”

The General Multiplication Rule Below shows the findings of recent study by school psychologists. What is the probability that a randomly selected student is Happy and a Boy? Boys Girls Happy 25 55 Sad 60 30

Conditional Probability Conditional probability takes into account a given condition. Ex. What is the probability that you will die of lung cancer if you smoke? Ex. What is the probability that you play soccer if you are a girl? Ex. What is the probability that you are a girl if you play soccer?

Conditional Probability Below is written the conditional probability “rule” It is read: the probability of B, given A” Remember that: if A and B are independent Helpful to draw contingency tables for these problems

Independent How do we know if two events are independent? (will help with independence assumption) Basically says, the probabilities of independent events don’t change when you find out that one of them has occurred.

Independence ≠ Disjoint The events are disjoint or mutually exclusive. They have no outcomes in common. Therefore, the outcome of one must change the probability of the other occurring.

Independence ≠ Disjoint Mutually exclusive events can’t be independent. They have no outcomes in common, so knowing that one occurred means that other didn’t. A common error is to treat disjoint events as if they were independent and apply the multiplication rule for independent events.

Beware of “without replacement” problems Ex. What is the chance that you are dealt five straight red cards in a game of poker?

Use of Tree Diagrams Very useful for problems dealing with conditional probabilities

Tree Diagrams A company has designed a test to determine if people are afflicted with a disease known as Ignoramusosis. In the general population, this disease strikes about 2% of the population. This test is highly accurate, able to correctly identify the disease 95% of the time. On the other hand, it is likely to give a false positive reading about 3% of the time. What is the probability that if you test positive, you actually have the disease?

Bayes’s Theorem NOT USED ON AP EXAM!! Additional information