Question

10. Adam and Beth are two salespeople at a car dealership. Their boss decides to give a bonus to whoever sells the next car. They will alternate serving customers. Thus, only one of them is serving a customer at any given time. Suppose there is an 18% chance Adam makes a sale, a 25% chance Beth makes a sale, and all customers are independent from one another. They will flip a fair coin to see who gets the first customer. What is the probability Beth gets the bonus?

Answer 1

Let A denotes that Adam gets the sales and Ac denotes that Adam didn't get the sale and similar for Beth is B and Bc. Then,

P(Beth gets the bonus)

= P(Beth gets to sale first)

= P(Ac B) + P(Ac Bc Ac B) + P(Ac Bc Ac Bc Ac B) + .........

= (0.82)*(0.25) + (0.82)*(0.75)*(0.82)*(0.25) + (0.82)*(0.75)*(0.82)*(0.75)*(0.82)*(0.25) + .....

= (0.82)*(0.25) [1 + 0.615 +0.615² +.... ]

= 0.205* 1 / 1 - 0.615

= 0.205 / 0.385

= 0.5325 [Rounded off to 4 decimal places]

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