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).

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warylucknow warylucknow · Answer 1 · 2020-02-06T12:53:29+01:00

Answer:

Step-by-step explanation:

Given:

[tex]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[/tex]

The rule of total probability states that:

[tex]P(A) = P(A\cap B) + P(A\cap B^{c})[/tex]

Compute the individual probabilities as follows:

[tex]P(S) = P(S\cap F) + P(S\cap F^{c})\\=0.44+0.13\\0.57[/tex]

[tex]P(S^{c}) = 1 - P(S)\\=1-0.57\\=0.43[/tex]

[tex]P(F) = P(S\cap F) + P(S^{c}\cap F)\\=0.44+0.11\\=0.55[/tex]

[tex]P(F^{c})=1-P(F)\\=1-0.55\\=0.45[/tex]

Conditional probability of an event A given B is:

[tex]P(A|B)=\frac{P(A\cap B)}{P(B)}[/tex]

[tex]P(S|F)=\frac{P(S\cap F)}{P(F)}\\=\frac{0.44}{0.55}\\=0.80[/tex]

[tex]P(S|F^{c})=\frac{P(S\cap F^{c})}{P(F^{c})}\\=\frac{0.13}{0.45}\\=0.289[/tex]

[tex]P(S^{c}|F)=\frac{P(S^{c}\cap F)}{P(F}\\=\frac{0.11}{0.55}\\=0.20[/tex]

[tex]P(S^{c}|F^{c})=\frac{P(S^{c}\cap F^{c})}{P(F^{c})}\\=\frac{0.32}{0.45}\\=0.711[/tex]