Chapter 15 Probability Rules!

Topics Probability Terminology ctd. Venn Diagrams Conditional Probability Tree Diagrams

Terminology Sample Space: The set of all possible outcomes. Event: one of the possible outcomes or combination of outcomes.

Example For each of the following, list the sample space and tell whether you think the events are equally likely. Roll two dice; record the sum of the numbers. A family has 3 children; record the genders in order of birth. Toss four coins; record the number of tails. Toss a coin 10 times; record the longest run of heads.

Venn Diagrams A B C

Venn Diagrams ctd. In the previous Venn Diagram, identify the following: A and B A or B CC A and B and C

Generalized Addition Rule from Venn Diagrams P(A or B) = P(A) + P(B) – P(A and B) This is known as the General Addition Rule Question: If A and B are disjoint (mutually exclusive), what is P(A and B)? P(A or B but not both) = P(A or B or C) =

Venn Diagram Example #1 A card is randomly drawn from a deck of 52 cards. What is the probability the card chosen is a 5 or a heart?

Venn Diagram Example #2 At a law firm, 80% of the attorneys have an administrative aide, 30% have an intern, and 25% have both an administrative aide and an intern. What percentage of the attorneys have neither?

Conditional Probability Recall the concept of contingency tables.

Conditional Probability What is the probability of randomly selecting a girl from the prior sample? What is the probability of randomly selecting a girl who prefers dogs? What is the probability that a randomly selected student prefers dogs, given that we have selected a girl?

Conditional Probability P(B|A) means the probability of B, given A.

Example ctd. What is the probability that a randomly selected student is a boy, given that he prefers cats?

Disjoint and Independent If P(A and B) = 0, then A and B are disjoint. If P(B|A) = P(B), then A and B are independent.

Tree Diagrams Golf example ctd. Assume the following probabilities for a good golfer. On the tee shot, you have a 60% chance of putting the ball on the green. If the ball does not land on the green, you have a 90% chance of each subsequent shot not from off the green landing on the green. Once the ball lands on the green, you have a 40% chance of making the first putt and a 90% chance of making each subsequent putt. What is the probability of shooting a 3 on the hole?

Reversing the Conditioning What if we know P(B|A), but we want to know P(A|B)? Suppose a politician proposed wide-spread, random cancer screening as a method of prevention. Assume the test is 98% effective. Now, assume that you were randomly chosen and took the test. It came back positive. What is the actual probability that you have cancer?

Assumptions Test effectiveness: 98% accurate (if someone has cancer, it will be positive 98% of the time and if someone does not have cancer, it will be negative 98% of the time). Actual cancer rate: 0.5% Take a guess—What do you think the probability you have cancer is, given you have been randomly tested and test positive for cancer?

Bayes Formula Bayes Rule for reversing conditioning. All the evidence you need to use a tree diagram when possible