Question

I need an example of a complex conditional probability question. With the question and answer..

Thanks!

Answer 1

Answer:-

The probability of an event occurring given that another event has already occurred is called a conditional probability.

Recall that when two events, A and B, are dependent, the probability of both occurring is

P(A and B) = P(A) × P(B given A)
or P(A and B) = P(A) × P(B | A)

Example:-

A box contains three coins: two regular coins and one fake two-headed coin (P(H)=1),

You pick a coin at random and toss it. What is the probability that it lands heads up?

You pick a coin at random and toss it, and get heads. What is the probability that it is the two-headed coin?

Solution

This is another typical problem for which the law of total probability is useful. Let C1 be the event that you choose a regular coin, and let C2 be the event that you choose the two-headed coin. Note that C1 and C2 form a partition of the sample space. We already know that

P(H|C1)=0.5,

P(H|C2)=1.

Thus, we can use the law of total probability to write

P(H) =P(H|C1)P(C1)+P(H|C2)P(C2)

=12.23+1.13

=23.

Now, for the second part of the problem.

P(C2|H) =P(H|C2)P(C2)P(H)

=1.1323

=12.