Question

Customers arriving to the food court in a university's student union are classified as either student, faculty member, or other. Let F denote that event that the next customer is a faculty member, and let O denote the event that the next customer is other. Suppose we begin observing the food court at some point in time Show that the probability that a faculty member arrives before an other is given by Pr(F)/ Pr(F) + Pr(O).

I know that I am supposed to apply the Law of Total Probability but since there are no values I'm at a loss with how to correctly answer this question.

Answer 1

Let S demote the event that the next customer is a student

Thus, P(S) + P(F) + P(O) = 1

For the event that a faculty member arrives before an other, the possibilities are F, SF, SSF, SSSF,....

e.g. SSF denotes the next customer is a student, again the next customer is a student, and then the next customer is a faculty member.

Thus, the required probability = P(F) + P(SF) + P(SSF) + ......

= P(F) + P(S)*P(F) + P(S)*P(S)*P(F) + ....... (Since the events are independent)

= P(F)*{1 + P(S) + P(S)^{2} + .......... }

This is sum of infinite GP the sum of which equals

So the probability that a faculty member arrives before an other

= =

=