Question

A study from a company reveals that 45% of the products that they sell have an excellent acceptance in the market, 35% have a moderate acceptance, and 20% have a poor acceptance in the market. The clients participate in the evaluation of new products during its period of promotion. In the past, 90% of the products that have an excellent acceptance in the market have received an evaluation of "A" during the promotion process. 50% of the products that are moderately accepted in the market have also received an evaluation of "A", and 7% of the products that have a poor acceptance in the market have also received an evaluation of "A".

A) What is the probability that a product achieves an evaluation of "A"?

B) If a new product has an evaluation of "A" during the promotion process, what is the probability of having an excellent acceptance in the market?

C) If a product doesn't achieve an evaluation of "A", what is the probability of having an excellent acceptance in the market?

Answer 1

Let us define the events first.

A: Received an evaluation during the promotion process.

A': Doesn't achieve an evaluation of A

E: Excellent acceptance in the market

M: Moderate acceptance in the market

P: Poor acceptance in the market.

P(E) = 0.45, P(M) = 0.35, P(P) = 0.20

90% of the products that have an excellent acceptance in the market have received an evaluation of "A" during the promotion process that is P(A | E) = 0.90

P(A | M) = 0.50 and P(A | P) = 0.07

(a) The probability that a product achieves an evaluation of "A" that is P(A)

The formula to find P(A) using total law of probability is,

P(A) = P(A | E) * P(E) + P(A | M) * P(M) + P(A | P) * P(P)

= 0.90 * 0.45 + 0.50 * 0.35 + 0.07 * 0.20

P(A) = 0.594

(b) If a new product has an evaluation of "A" during the promotion process, what is the probability of having an excellent acceptance in the market that is P(E | A)

Using Bayes theorem the formula to find the conditional probability P(E | A) is,

Therefore, If a new product has an evaluation of "A" during the promotion process, then the probability of having an excellent acceptance in the market is 0.6818

(c) If a product doesn't achieve an evaluation of "A", what is the probability of having an excellent acceptance in the market

that is P(E | A')

The formula to find the conditional probability is,

P(A' | E) = 1 - P(A | E) = 1 - 0.90 = 0.10

P(A') = 1 - P(A) = 1 - 0.594 = 0.406

Therefore, If a product doesn't achieve an evaluation of "A", then the probability of having an excellent acceptance in the market 0.1108