Math CentralQuandaries & Queries


Question from Vi, a student:

A microwave oven manufacturer claims that only 10% of the ovens it makes will need repair in the first year. Suppose three recent customers are independently chosen.
a. If the manufacturer is correct, what is the probability that at least two of the three ovens will need repair in the first year?
b. If at least two of the three customers' ovens need repair in the first year, what inference may be made about the manufacturer's claim?

I know the answer to (a) is .028, but I don't understand how one gets there.

Hi Vi,

You can approach this problem using the binomial distribution but I prefer to describe it using a tree diagram.

tree diagram

Each level in the tree represents the response from a customer, R if their oven needs repair and N if it does not need repair. If the manufacturer is correct then the first customer responds with R or N with probabilities 0.1 and 0.9 respectively. Regardless of the response from the first customer the second customer can respond with R or N again with probabilities 0.1 and 0.9, and likewise for the third customer.

Each path through the tree from bottom to top then represents a possible sequence of responses. For example the path RRR represents three rejections and the probability that you receive this response is 0.1 × 0.1 ×0.1 = 0.001. Now look for paths through the tree with two Rs and one N. There are three such paths, RRN, RNR and NRR. Each of these paths has an associated probability of 0.1 × 0.1 × 0.9 = 0.009. Thus the probability that the responses from the three customers contain exactly 2 rs is 3 × 0.009 = 0.027.

Thus the probability that at least two of the three ovens will need repair in the first year is 0.001 + 0.027 = 0.028.


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