Question 5.1: Car Hire Request Study The Zeplin car hire company has a fle......

The random variable (i.e. the number of hire requests for an Opel) is discrete, since 0, 1, 2, 3, 4, etc. Opels can be requested for hire on a given day. For this discrete random variable to follow the binomial process, it must satisfy the four conditions defined above.

Condition 1 is satisfied, since the random variable is observed five times (i.e. a sample of five hiring requests was studied). Hence n = 5. Each of the five reservation requests is a single trial (or object) in the study of car hire request patterns.

Condition 2 is satisfied, since there are only two possible outcomes on each client request:

• A client requests the hire of an Opel (success outcome).
• A client requests the hire of another make of car, i.e. not an Opel (failure outcome).

Condition 3 is satisfied, since the probability of the success outcome is constant and is derived from the statement that ‘experience has shown that one in four clients request to hire an Opel’.

Thus p (= probability of a client requesting to hire an Opel) = 0.25

(1 − p) (= probability of a client requesting another make of car) = 0.75

Condition 4 is satisfied, since the trials are independent. Each client’s car preference request is independent of every other client’s car preference request. This implies that p will not change from one client request to another.

Since all the conditions for the binomial process have been satisfied, the binomial question can be addressed: ‘What is the probability that two out of five clients will request to hire an Opel?’

Find: P(x = 2) when n = 5 and p = 0.25.

Then: P(x = 2) = _{5}C_{2}(1-0.25)^{5-2} = (10)(0.0625)(0.4219) = 0.264

Thus there is a 26.4% chance that two out of five randomly selected clients will request an Opel.