Elementary Statistics – More Probability Problems

 

  1. Students sometimes confuse the ideas of disjointness and independence. Recall two events are disjoint if they share no outcomes in common, and two events are independent if knowing one event occurs does not change the probability the other occurs.

 

    1. Suppose two events, A and B, (each with positive probability) are disjoint (for example, rolling an even number on a 6-sided die and rolling an odd number on a 6-sided die). Determine P(A|B). Are A and B independent events?

 

 

 

    1. At a local college, 52% of students are female, 60% of students are from Wisconsin, and 15% are English majors. Furthermore, 31.2% are female students from Wisconsin, and 10% are female English majors. (You can think of these percentages as probabilities.) Are the events {female} and {Wisconsin native} independent? Are the events {female} and {English major} independent?

 

 

 

    1. Note part a shows that disjoint events must be dependent, and part b shows that non-disjoint events can be either independent or dependent.

 

 

  1. What happens when outcomes in a sample space are not equally likely? Suppose an unfair coin is flipped three consecutive times, and each time the upward face is recorded. For this coin, P(heads) = 0.7 and P(tails) = 0.3.

 

    1. Write out the sample space for this experiment.

 

 

 

    1. Let A = {tail on the first flip} and B = {exactly two tails in the three flips}. Determine P(B|A). (Note this conditional probability is not 0.5, which is what it would be if the sample space outcomes were equally likely. Furthermore, this conditional probability is not 0.189—if you got this incorrect answer, you erroneously assumed A and B were independent when computing P(A and B).)

 

 

 

    1. Are the events A and B (defined in part b) independent? (In this example, the coin flips are independent, but the compound events A and B are not independent.)

 

 

 

  1. A health study tracked a group of people for five years. At the beginning of the study, 35% were classified as smokers and 65% were classified as nonsmokers. Results of the study showed that smokers were twice as likely to die as nonsmokers during the five-year study. Given that a randomly selected participant dies over the five-year period, determine the conditional probability the participant was a smoker. (Hint: Use a tree diagram.)