A consumer products company found that 42% of successful products also received favorable results from test market research, whereas 11% had unfavorable results but nevertheless were successful. That is, P(successful product and favorable test market)equals0.42 and P(successful product and unfavorable test market)equals0.11. They also found that 35% of unsuccessful products had unfavorable research results, whereas 12% of them had favorable research results, that is P(unsuccessful product and unfavorable test market)equals0.35 and P(unsuccessful product and favorable test market)equals0.12. Find the probabilities of successful and unsuccessful products given known test market results, that is, P(successful product given favorable test market), P(successful product given unfavorable test market), P(unsuccessful product given favorable test market), and P(unsuccessful product given unfavorable test market).

(1) The probability of a successful product given the product is favorable is 0.7778.

(2) The probability of a successful product given the product is unfavorable is 0.2391.

(3) The probability of a unsuccessful product given the product is favorable is 0.2222.

(4) The probability of a unsuccessful product given the product is favorable is 0.7609.

Step-by-step explanation:

Denote the events as follows:

S = a product is successful.

F = a product is favorable.

The information provided is:

The law of total probability states that:

Use the law of total probability to compute the probability of a favorable product as follows:

The probability of a favorable product is 0.54.

The conditional probability of an event A given that another event B has already occurred is:

(1)

Compute the value of P (S|F) as follows:

Thus, the probability of a successful product given the product is favorable is 0.7778.

(2)

Compute the value of as follows:

Thus, the probability of a successful product given the product is unfavorable is 0.2391.

(3)

Compute the value of as follows:

Thus, the probability of a unsuccessful product given the product is favorable is 0.2222.

(4)

Compute the value of as follows:

Thus, the probability of a unsuccessful product given the product is favorable is 0.7609.