Let's approach this problem step-by-step.
(a) Among fatal crashes in normal weather, what is the probability that a randomly selected fatal crash occurs when it is dark, but lighted?
To find this probability, we need to use the data given for fatal crashes in normal weather. The steps to solve this are as follows:
1. Identify the relevant data:
- Normal weather crashes that occurred in different light conditions:
- Daylight: [tex]\(14307\)[/tex]
- Dark, but lighted: [tex]\(5875\)[/tex]
- Dark: [tex]\(8151\)[/tex]
- Dawn/Dusk: [tex]\(1183\)[/tex]
- Other: [tex]\(65\)[/tex]
2. Calculate the total number of fatal crashes in normal weather:
[tex]\[
\text{Total normal weather crashes} = 14307 + 5875 + 8151 + 1183 + 65 = 29581
\][/tex]
3. Determine the number of fatal crashes in normal weather that occurred when it was dark but lighted:
- This number is [tex]\(5875\)[/tex].
4. Compute the probability:
[tex]\[
\text{Probability} = \frac{\text{Number of dark but lighted crashes}}{\text{Total normal weather crashes}} = \frac{5875}{29581}
\][/tex]
5. Round to three decimal places:
[tex]\[
\frac{5875}{29581} \approx 0.199
\][/tex]
So, the probability that a randomly selected fatal crash in normal weather occurs when it is dark, but lighted is approximately [tex]\(0.199\)[/tex].
(b) Among fatal crashes when it is dark, but lighted, what is the probability that a randomly selected fatal crash occurs in normal weather?
To find this probability, we need to look at the total number of fatal crashes that occurred in all weather conditions when it was dark but lighted, and then determine what proportion of these occurred in normal weather. The steps to solve this are as follows:
1. Identify the number of dark but lighted crashes for each weather condition:
- Normal: [tex]\(5875\)[/tex]
- Rain: [tex]\(497\)[/tex]
- Snow/Sleet: [tex]\(51\)[/tex]
- Other: [tex]\(54\)[/tex]
- Unknown: [tex]\(255\)[/tex]
2. Calculate the total number of fatal crashes that occurred when it was dark but lighted, across all weather conditions:
[tex]\[
\text{Total dark but lighted crashes} = 5875 + 497 + 51 + 54 + 255 = 6732
\][/tex]
3. Determine the number of dark but lighted crashes that happened in normal weather:
- This number is [tex]\(5875\)[/tex].
4. Compute the probability:
[tex]\[
\text{Probability} = \frac{\text{Number of dark but lighted crashes in normal weather}}{\text{Total dark but lighted crashes}} = \frac{5875}{6732}
\][/tex]
5. Round to three decimal places:
[tex]\[
\frac{5875}{6732} \approx 0.873
\][/tex]
So, the probability that a randomly selected fatal crash when it is dark, but lighted occurs in normal weather is approximately [tex]\(0.873\)[/tex].
Final answers:
- (a) 0.199
- (b) 0.873
