Question

1. Explain why, if “A” stands for “There is thunder” and “B” stands for “It is raining,” you would expect p(A|B) to be greater than P(A).
2. Suppose you have a fair coin which is equally likely to come up heads or tails. Let “A” stand for “the first toss comes up heads” and “B” stands for the 2^nd toss comes up heads.” Calculate the following:
3. P(A)=
4. P(B)=
5. P(A&B)=
6. P(B|A)=

Answer 1

Thunder and rain-

From the definition of conditional probability,

Further we have,

The event denotes 'thunder without raining' and in general, this is very much rear event.

As it does not rain always, .

Hence, in general P(A|B) is expected to be greater than P(A).

Heads and tails-

A denotes the first toss coming up head and B denotes that in the second toss.

Clearly, all tosses being independent from other tosses events A and B are independent.

Required probabilities are as follows.