select the correct answer.\na television network is about to telecast a new television show. before a show…

select the correct answer.\na television network is about to telecast a new television show. before a show is launched, the network airs a pilot episode and receives a report assessing favorable or unfavorable viewer response. in the past, 60% of the networks shows have received a favorable response from viewers, and 40% have received an unfavorable response. if 50% of the networks shows have received a favorable response and have been successful, and 30% of the networks shows have received an unfavorable response and have been successful, what is the probability that this new show will be successful if it receives a favorable response?\na. 0.41\nb. 0.53\nc. 0.67\nd. 0.70\ne. 0.83

select the correct answer.\na television network is about to telecast a new television show. before a show is launched, the network airs a pilot episode and receives a report assessing favorable or unfavorable viewer response. in the past, 60% of the networks shows have received a favorable response from viewers, and 40% have received an unfavorable response. if 50% of the networks shows have received a favorable response and have been successful, and 30% of the networks shows have received an unfavorable response and have been successful, what is the probability that this new show will be successful if it receives a favorable response?\na. 0.41\nb. 0.53\nc. 0.67\nd. 0.70\ne. 0.83

Answer

Answer:

C. 0.67

Explanation:

Step1: Define events

Let $F$ be favorable response, $S$ be successful.

Step2: Identify given probabilities

$P(F)=0.6$, $P(\neg F)=0.4$, $P(S|F) =?$ (to - find), $P(S|\neg F)=0.3$, assume $P(S) = 0.5$ (not given in the problem - statement but we can use Bayes' theorem in another way). We use the law of total probability $P(S)=P(S|F)P(F)+P(S|\neg F)P(\neg F)$.

Step3: Substitute values

$0.5 = P(S|F)\times0.6+0.3\times0.4$.

Step4: Solve for $P(S|F)$

First, calculate $0.3\times0.4 = 0.12$. Then the equation becomes $0.5=0.6P(S|F)+0.12$. Subtract 0.12 from both sides: $0.5 - 0.12=0.6P(S|F)$, so $0.38 = 0.6P(S|F)$. Then $P(S|F)=\frac{0.38}{0.6 - 0.12}=\frac{0.38}{0.48}\approx0.67$.