Question

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).

Answer 1

Answer:

(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:

$P(S\cap F)=0.42\\P(S\cap F^{c})=0.11\\P(S^{c}\cap F)=0.12\\P(S^{c}\cap F^{c})=0.35$

The law of total probability states that:

$P(A)=P(A\cap B)+P(A\cap B^{c})$

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

$P(F)=P(S\cap F)+P(S^{c}\cap F)\\=0.42+0.12\\=0.54$

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:

$P(A|B)=\frac{P(A\cap B)}{P(B)}$

(1)

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

$P(S|F)=\frac{P(S\cap F)}{P(F)}=\frac{0.42}{0.54}=0.7778$

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

(2)

Compute the value of $P(S|F^{c})$ as follows:

$P(S|F^{c})=\frac{P(S\cap F^{c})}{P(F^{c})}=\frac{0.11}{(1-0.54)}=0.2391$

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

(3)

Compute the value of $P (S^{c}|F)$ as follows:

$P (S^{c}|F)=\frac{P(S^{c}\cap F)}{P(F)}=\frac{0.12}{0.54}=0.2222$

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

(4)

Compute the value of $P (S^{c}|F^{c})$ as follows:

$P (S^{c}|F)=\frac{P(S^{c}\cap F^{c})}{P(F^{c})}=\frac{0.35}{(1-0.54)}=0.7609$

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