Let us take an example:

Ex: consider the experiment of tossing a coin.if the coin shows head,toss it again but if it shows the tail, then throw a die.

find the conditional probability of the event that the die shows a number greater than 4 given that there is at least one tail.

Solution:

Given, the experiment of tossing a coin. If the coin shows head, toss it again but if it shows a tails, then throw a die.

Now, the tree diagram is constructed as follows:

Now, the sample space of the experiment is:

S = {(H,H), (H,T), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}

Here (H, H) denotes that both the tosses result into head and

(T, i) denotes that the first toss result into a tail and the number i appeared on the dice for i = 1 to 8

Now the probability of these 8 elementary events are

1/4, 1/4, 1/12, 1/12, 1/12, 1/12, 1/12, 1/12 respectively.

Let F denotes the event that there is at least one tail

and E denotes the event that the dice shows a number greater than 4

Now, F = {(H,T), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}

and E = {(T,5), (T,6)}

and E ∩ F = {(T,5), (T,6)}

Now, P(F) = P{(H,T)}+ P{(T,1)} + P{(T,2)} + P{(T,3)} + P{(T,4)} + P{(T,5)} + P{(T,6)}

              = 1/4 + 1/12 + 1/12 + 1/12 + 1/12 + 1/12 + 1/12

              = 1/4 + 6/12

              = 1/4 + 1/2

              = (2 + 1)/4

              = 3/4

and P(E ∩ F) = P{(T,5)}, P{(T,6)}

                  = 1/12 + 1/12

                  = 2/12

                  = 1/6