Find the Probability

Question

Ian, Tyrone, Sergio and Larry have all been invited to a dinner party. They arrive randomly and each person arrives at a different time.

 

In how many ways can they arrive?

In how many ways can Ian arrive first and Larry last?

Find the probability that Ian will arrive first and Larry last?

Answer 1

Hello Kathy,

 

In how many different ways can they arrive?

 

For the first to arrive, there are four possible choices. Whomever arrives first, there are then three possible choices for each of those four possible firsts, so 12 different ways the first two can arrive. But whomever arrives first and second, there will be two possible choices for third, so 24 possible different first 3 people. Finally one possible choice for last for each of the 24 possible first three arrivals. 

So the answer is 4x3x2x1 = 24 different ways they can arrive.

 

In how many ways can Ian arrive first and Larry last?

I am going to assume that this means the event Ian arrives first and Larry last in the same event, rather than the events where Ian arrives first or Larry arrives last. If Ian is first and Larry is last then either Sergio is second and Tyrone third or Tyrone is second and Sergio is third, so two possible ways this can happen.

 

Find the probability that Ian arrives first and Larry last.

That they arrive randomly means that each of the 24 different orderings of their arrival is equally likely. Since the last part showed that two of the orderings have Ian first and Larry last, the probability of this is 2/24, or 1/12, or about 0.083.

 

Hope this helps, let me know if you have questions.

 

Kind regards,

Hugh