Math 2 HonorsName______

Lesson 4-4 Conditional Probability

Learning Goals:

  • I can define and/or identify the following: event, sample space, union, intersection, complement, independent event, dependent event and conditional probability.
  • I can illustrate the concept of conditional probability using everyday examples of dependent events.
  • I can explain that A and B are independent events if the occurrence of A does not impact the probability of B occurring and vice verse (i.e., A and B are independent events if P(B|A)=P(B) and P(A|B)=P(A)).
  • I can calculate,interpret and explain the conditional probability of A given B using the model .

1.Count the number of students in your classroom who are wearing tennis shoes. Count the number of girls. Count the number of students who are wearing tennis shoes and are girls. Record the number of students who fall into each category in the following table.

Wearing Tennis Shoes / Not Wearing Tennis Shoes / Total
Boy
Girl
Total
  1. Suppose you select a student at random from your class. What is the probability that the student is wearing tennis shoes?
  1. Suppose you select a student at random from your class. What is the probability that the student is a girl?
  1. Does using the Multiplication Rule correctly compute the probability that the student is wearing tennis shoes and is a girl?

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  1. How is this situation different from previous situations in which the Multiplication Rule gave the correct probability?

2.The phrase “the probability event A occurs given that event B occurs” is written symbolically as This conditional probability sometimes is read as “the probability of A given B.” The table below categorizes the preferences of 300 students in a junior class about plans for their prom.

Preference for Location
Hotel / Rec Center
Preference for Band / Hip-Hop / 73 / 80
Classic Rock / 55 / 92

Suppose you pick a student at random from this class. Find each of the following probabilities.

  1. P(prefers hotel)
  2. P(prefers hip-hop band)
  3. P(prefers hotel and prefers hip-hop band)
  4. P(prefers hotel or prefers hip-hop band)
  5. P(prefers hotel | prefers hip-hop band)
  6. P(prefers hip-hop band | prefers hotel)

3.Recall that events A and B are independent if knowing whether one of the events occurs does not change the probability that the other event occurs.

  1. Using the data from Problem 1, suppose you pick a student at random. Find P(wearing tennis shoes | is a girl). How does this compare to P(wearing tennis shoes)?
  1. Are the events wearing tennis shoes and is a girl independent? Why or why not?
  1. Consider the table from a different class.

Wearing Tennis Shoes / Not Wearing Tennis Shoes
Boy / 5 / 9
Girl / 10 / 18

Suppose you pick a student at random from this class.

  1. Find P(wearing tennis shoes)
  2. Find P(wearing tennis shoes | is a girl)
  3. Are the events wearing tennis shoes and is a girl independent? Why or why not?
  1. If events A and B are independent, how are P(A) and P(A | B) related?

4.Suppose that you roll a pair of dice.

  1. Which is greater? P(doubles) or P(doubles | sum of 2)?
  1. Are the events getting doubles and getting a sum of 2 independent? If not, how would you describe the relationship?

5.Refer to the table in Problem 2.

  1. If you select a junior at random, are the events prefers hotel and prefers hip-hop band independent? Explain.
  1. If you select a junior at random, are the events prefers hotel and prefers hip-hop band mutually exclusive? Explain.
  1. Are the events prefers hotel and prefers rec center mutually exclusive? Explain.

6.Suppose you pick a high school student at random. For each of the pairs of events in Parts a, b, and c, write the mathematical equality or inequality that applies:

P(A) = P(A | B), P(A) > P(A | B),or P(A) < P(A | B).

  1. A is the event that the student is male, and B is the event that the student is over six feet tall.
  1. A is the event that the student is female, and B is the event that the student has brown eyes.
  1. A is the event that the student is a member of the French club, and B is the event that the student is taking a French class.
  1. Which of the pairs of events is it safe to assume are independent? Explain your reasoning.