Prisoners

It is estimated that 10% of all federal prisoners have a positive self-image, 40% have a neutral self-image, and the rest have a negative self-image. The estimated probability of rehabilitating a prisoner with a negative self-image is 0.1. With a neutral self-image, this estimated probability is 0.4, and with a positive one, it is 0.8.

A prisoner is rehabilitated. What is the probability that the original self-image was negative?
A) 0.17 C) 0.63
B) 0.71 D) 0.36

A prisoner is not rehabilitated. What is the probability that the original self-image was negative?
A) 0.36 C) 0.63
B) 0.71 D) 0.17

Correct answer:

p₁ =  0.1724
p₂ =  0.6338

Step-by-step explanation:

$$\begin{gathered} r=0.10\cdot 0.8+0.40\cdot 0.4+0.50\cdot 0.10=0.29 \\ p(A|B)\cdot p(B)=p(A\cap B) \\ {p}_1=0.50\cdot 0.1/r=0.50\cdot 0.1/0.29=0.1724 \end{gathered}$$

$$\begin{gathered} n=1-r=1-0.29=\frac{71}{100}=0.71 \\ {p}_2=0.50\cdot (1-0.1)/n=0.50\cdot (1-0.1)/0.71=0.6338 \end{gathered}$$

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