Question

The first baseman in a baseball game is thrown two balls during an inning. Define A = {catches both balls}, B = {catches at least one ball}, C = {misses both balls}. Which of the following is false?
1. A and C are disjoint.
2. B and C are disjoint.
3. A and B are independent.
4. P(A) + P(B) + P(C) ≠ 1.

You flip a coin x number of times and calculate the probability of heads as (number of heads)/x. What happens to this probability as x gets larger?
1. It gets smaller
2. It gets larger
3. It stays the same (0.5)
4. It gets closer and closer to 0.50

Answer 1

a) The false statement is:

3. A and B are independent.

If A and B are independent, P(B | A) = P(B)

Here, P(B | A) = 1 because given both balls are caught, its certain that at least 1 ball is caught. But, P(B) need not be 1. Therefore P(B | A) P(B) and the events are dependent.

b) As x gets larger,

4. It gets closer and closer to 0.50

According to law of large numbers, the sample proportion gets closer to the populatio proportion as the number of samples increases. We know that for a fair coin, the population proportion must be 0.5. Therefore, as x gets larger, P(heads) gets closer to 0.5.