Question

A soft-drink machine dispenses only regular Coke and Diet Coke. Sixty percent of all purchases from this machine are diet drinks. The machine currently has 10 cans of each type. If 15 customers want to purchase drinks before the machine is restocked, what is the probability that each of the 15 is able to purchase the type of drink desired? (Hint: Let \(x\) denote the number among the 15 who want a diet drink. For which possible values of \(x\) is everyone satisfied?)

Short answer

The probability that each of the 15 customers is able to purchase the type of drink desired is calculated by summing the probabilities for each possible value of \(x\) from 0 to 10.

Step-by-step solution

Identify The Variable Range

Based on the hint given, define \(x\) as the number of customers who want diet drinks. As there are 15 customers and 10 cans of each drink type, \(x\) can range from 0 to 10. If \(x\) is less than 0 or greater than 10, there would not be enough cans for everyone.

Calculate The Total Possibilities

Calculate the total number of possibilities. This can be done by applying the binomial theorem for \(x\) number of customers who want diet drinks and \(15-x\) who want regular drinks. So, the total possibilities are \(\binom{15}{x}\) for diet Coke and \(\binom{15}{15-x}\) for regular Coke.

Calculate Probability For Each \(x\)

The probability that \(x\) customers desire a diet drink is \(0.6^{x}\) and the probability that \(15-x\) customers want a regular drink is \(0.4^{15-x}\). Simplifying, the probability for each \(x\) is given by \(\binom{15}{x}\) * \(0.6^{x}\) * \(0.4^{15-x}\).

Sum Up Probabilities

Sum up all the probabilities calculated in the previous step for \(x\) ranging from 0 to 10. Summing these probabilities will give the total probability that each of the 15 customers is able to purchase the type of drink desired.