Probability and Bayes Theorem

Question

I have been trying to solve this problem using Bayes Theorem all week with no luck.

A study showed that in​ 1990, $45​\%$ of all those involved in a fatal car crash wore seat belts. Of those in a fatal crash who wore seat​ belts, $43\%$ were injured and $27\%$ were killed. For those not wearing seat​ belts, the comparable figures were $42​\%$ and $49​\%$, respectively.

Find the probability that a randomly selected person who was unharmed in a fatal crash was not wearing a seat belt.

My initial assumption was they are asking for P(no seat belt|not injured) and P(no seat belt|not killed) but that does not seem to be the correct answer.

My next thought was that they are asking for the following P(no seat belt|not injured and not killed)

My problem being I'm unsure of how to get given portion of the probability. Any ideas would be appreciated.

Answer 1

Well, the first thing we usually do in such cases is to name the events. So, let $B=\{$ one passenger wears seat belt during a fatal crash$\}$, $K=\{$one passenger gets killed during a fatal crash$\}$, $I=\{$one passenger gets injured during a fatal crash$\}$ and $U=\{$ one passenger is unharmed during a fatal crash $\}$.

What we know is $P(B) =0.45$, $P(I\mid B) = 0.43$ and $P(K\mid B) = 0.27$. The last 2 imply $P(U\mid B) = 1-0.43-0.27 = 0.3$

Also, we know $P(I\mid B') = 0.42$ and $P(K\mid B') = 0.49$. Again, we have that $P(U\mid B') = 1-0.42-0.49 = 0.09$.

The probability we are looking for is $\boxed{P\big(B' \mid U \big)}.$

We have that $$P(U)\begin{array}[t]{l}= P(B)\cdot P\big(U\mid B\big)+ P(B')\cdot P\big(U\mid B'\big)\\ =0.45\times 0.3+(1-0.45)\times 0.09\\ =0.1845. \end{array}$$

Also, we have that $$P\big(B'\mid U\big) = \dfrac{ P\big(U\mid B')\cdot P(B')}{P(U)}=\dfrac{0.0495}{0.1845}\approx 0.268.$$

Answer 2

My interpretation is that the question asks for $P(S^c|U)$, where $S$ means "wore seatbelt" and $U$ means neither injured or killed. Then

$$P(S^c|U) = \frac{P(S^c U)}{P(U)} = \frac{P(S^c U)}{P(SU) + P(S^c U)},$$

where intersections are denoted as products. This is in the exact format of Bayes' Theorem. You ought to be able to get both of the probabilities on the RHS of this equation.

For example, $P(SU) = P(S)P(U|S) = (.45)(1 - .27 - .43).$ You ought to be able to take it from there.

It would have been nice of you to share what you believe is the correct answer.