# Probability

1. Show that if Pr(A|B) = Pr(A|Bc) then Pr(Ac|B) = Pr(Ac|Bc).
This means that since if Pr(A|B) = Pr(A|Bc) is true only if A and B are independent we also have Pr(Ac|B) = Pr(Ac|Bc) is true only if A and B are independent.
2. Show that if Pr(A|B) = Pr(A) then Pr(Ac|B) = Pr(Ac). This means that since if Pr(A|B) = Pr(A) is true only if A and B are independent we also have Pr(Ac|B) = Pr(Ac) is only true if A and B are independent.
3. Suppose we roll a six-sided die and define the following events. A = get an even number, so get 2 or 4 or 6
B = get a number less than 4, so get 1 or 2 or 3 Are A and B independent? Justify your answer.
4. Suppose we roll a six-sided die and define the following events. A = get an even number, so get 2 or 4 or 6
B = get a number less than 3, so get 1 or 2
5. Suppose that 1/3 of new businesses fail in one year. Of those remaining after one year, 1/4 fail in their second year.
1. (a) What is the probability that a new business will survive for two years?
2. (b) What is the probability that a new business will fail in its second year? Hint: To fail in the second means you have to have survived the first.
6. Suppose events A and B are not independent. For each of the following, say if the statement is true or false. (a) Pr(Ac|B) ̸= Pr(Ac); (b) Pr(A|Bc) ̸= Pr(A); (c) Pr(Ac|Bc) ̸= Pr(Ac); (d) Pr(A and B) = Pr(A)Pr(B)
7. A box contains 4 coins coin 1 has both sides tails
coin 2 has both sides heads
coin 3 has both sides heads
coin 4 is a regular coin (1 side head, other side tails)
1. (a) If we randomly choose one coin from the box and flip, what is the probability we get heads?
2. (b) Suppose we randomly choose a coin, flip, and get heads. What is the probability that the coin that was chosen is the regular (1 side head, other side tails) coin?

2

1. In a recent school year in the state of Washington, there were 326,000 high school students. Of these, 159,000 were girls and 167,000 were boys. Among the girls, 7,800 dropped out of school, and among the boys, 10,300 dropped out. A student is chosen at random. Round all answers to 4 digits past the decimal point. (a) What is the probability that the student is male? (b) What is the probability that the student dropped out? (c) What is the probability that the student is male and dropped out? (d) Given that the student is male, what is the probability that he dropped out? (e) Given that the student dropped out, what is the probability that the student is male?
2. Suppose that 1% of the population have a disease called stataphobia. There is a diagnostic test with 0.95 probability of being positive when a person has stataphobia and 0.8 probability of being negative when a person does not have stataphobia. If a person tests positive for stataphobia, what is the probability they have stataphobia? step 1 define notation
D = having stataphobia
T = testing positive for stataphobia step 2 Identify probabilities given.
There are three probabilities given in the question (0.01, 0.95, and 0.8). Which probabilities are these? step 3 Identify the probability you are asked to find. What probability is the question asking you to find? step 4 Use probability rules to find the probability you identified in step 3.