Question

A consumer products company found that 44​% of successful products also received favorable results from test market​ research, whereas 13​% had unfavorable results but nevertheless were successful. That​ is, P(successful product and favorable test ​market) = 0.44 and​ P(successful product and unfavorable test ​market) = 0.13. They also found that 32​% of unsuccessful products had unfavorable research​ results, whereas 11​% of them had favorable research​ results, that is​ P(unsuccessful product and unfavorable test ​market) = 0.32 and​ P(unsuccessful product and favorable test ​market) = 0.11.
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. 0.80
2. 0.289
3. 0.20
4. 0.711

Step-by-step explanation:

Given:

$P(S\cap F)=0.44\\P(S\cap F^{c})=0.13\\P(S^{c}\cap F^{c}) = 0.32\\P(S^{c}\cap F) = 0.11$

The rule of total probability states that:

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

Compute the individual probabilities as follows:

$P(S) = P(S\cap F) + P(S\cap F^{c})\\=0.44+0.13\\0.57$

$P(S^{c}) = 1 - P(S)\\=1-0.57\\=0.43$

$P(F) = P(S\cap F) + P(S^{c}\cap F)\\=0.44+0.11\\=0.55$

$P(F^{c})=1-P(F)\\=1-0.55\\=0.45$

Conditional probability of an event A given B is:

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

• Compute the value of $P(S|F)$:

         $P(S|F)=\frac{P(S\cap F)}{P(F)}\\=\frac{0.44}{0.55}\\=0.80$

• Compute the value of $P(S|F^{c})$

        $P(S|F^{c})=\frac{P(S\cap F^{c})}{P(F^{c})}\\=\frac{0.13}{0.45}\\=0.289$

• Compute the value of $P(S^{c}|F)$

        $P(S^{c}|F)=\frac{P(S^{c}\cap F)}{P(F}\\=\frac{0.11}{0.55}\\=0.20$

• Compute the value of$P(S^{c}|F^{c})$

       $P(S^{c}|F^{c})=\frac{P(S^{c}\cap F^{c})}{P(F^{c})}\\=\frac{0.32}{0.45}\\=0.711$