Independent and Dependent Events & Conditional Probability PDF

Topic: Independent and dependent events and conditional probability Question: Events A and B are independent events. Find P(B) if P(A and B) = 0.25 and P(A) = 0.5. Answer choices: A P(B) = 0.125 B P(B) = 0.45 C P(B) = 0.5 D Not enough information...

Topic: Independent and dependent events and conditional probability Question: Events A and B are independent events. Find P(B) if P(A and B) = 0.25 and P(A) = 0.5. Answer choices: A P(B) = 0.125 B P(B) = 0.45 C P(B) = 0.5 D Not enough information 133 Solution: C Since the events are independent, events we know that P(A and B) = P(A) ⋅ P(B) We can plug in P(A and B) = 0.25 and P(A) = 0.5 and solve for P(B). 0.25 = 0.5 ⋅ P(B) 0.25 P(B) = 0.5 P(B) = 0.5 134 Topic: Independent and dependent events and conditional probability Question: Events A and B are dependent events. If P(A and B) = 0.7 and P(B) = 0.875, what is P(A)? Answer choices: A P(A) = 0.875 B P(A) = 0.8 C P(A) = 0.6125 D Not enough information 135 Solution: D These events are dependent events, so we can say P(A and B) = P(A) ⋅ P(B | A) We know that P(A and B) = 0.7, but we would also need to know P(B | A) in order to be able to solve for P(A). Therefore, we don’t have enough information to solve the problem. 136 Topic: Independent and dependent events and conditional probability Question: Suppose that Katie rolls a six-sided die twice. Event A is that the first roll is a 6, so P(A) is the probability that the first roll is a 6. Event B is that the second roll is a 6, so P(B) is the probability that the second roll is a 6. Which statement is false? Answer choices: A The events are independent. B The events are dependent. C P(A and B) = P(A) ⋅ P(B | A) D P(A and B) = P(A) ⋅ P(B) 137 Solution: B Events can’t be independent and dependent at the same time, so either answer choice A is false or answer choice B is false. The rolls are independent if we can show that P(A and B) = P(A) ⋅ P(B). If events are independent, it doesn’t necessarily mean that P(A and B) = P(A) ⋅ P(B | A) is a false statement. It just means that P(B) = P(B | A). P(A) is the probability that the first die lands on 6, so P(A) = 1/6. P(B) is the probability that the second die lands on 6, so P(B) = 1/6. P(A and B) is the probability of rolling a 6 on both dice, so P(A and B) = 1/36. Now we can check for independence. P(A and B) = P(A) ⋅ P(B) 1 1 1 = ⋅ 36 6 6 1 1 = 36 36 Because this equation is true, the events are independent, not dependent. 138