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 total probability can be calculated by summing up the binomial probabilities for each possible value of \(x\) from 0 to 10, using a binomial probability formula. The specific answer depends on the calculated values.

Step-by-step solution

Identify the possible values for x

Given that the machine has 10 cans of each type of drink, in order for all customers to get their desired drink, the possible values of \(x\) (those who want diet drink) can range from 0 to 10.

Calculate binomial probabilities for each possible value of x

We need to calculate the binomial probabilities for the range of \(x\) using the formula for binomial distribution: \[ P(x) = \binom{n}{x} *(p^x) * ((1-p)^{n-x}) \] where n is the total number of trials (15 in this case), \(x\) is the number of successes, and \(p\) is the probability of success on any given trial (0.6 for diet drink in this case).

Calculate the total probability

Now, by adding up the calculated probabilities for each possible value of \(x\) from 0 to 10, we can obtain the total probability that each customer is able to purchase the type of drink desired.