What is the probability that the first person encountered with coulrophobia is the third person met in a random encounter?

To find the probability that the first person encountered with coulrophobia is the third person met, we can utilize the concept of geometric probability.

In this scenario, we assume that each person met has a consistent probability of having coulrophobia, denoted as ( p ). The probability that a person does not have coulrophobia would then be ( 1 - p ). For the first two people encountered not to have coulrophobia while the third person does, the situation can be modeled as follows:

1. The first person does not have coulrophobia: This occurs with probability ( 1 - p ).

2. The second person also does not have coulrophobia: This again occurs with probability ( 1 - p ).

3. The third person does have coulrophobia: This occurs with probability ( p ).

The probability of this sequence of events happening—the first and the second person not having coulrophobia, but the third one having it—is given by multiplying the probabilities of each event:

[ P(\text{first having coulrophobia at the third encounter}) = (1 - p)(1 - p)(p) = (1 - p)^2 p. ]

In this case, if we