Sure, let's break down each part of the problem step-by-step.
### Part (a)
Among fatal crashes in normal weather, we are interested in finding the probability that a randomly selected fatal crash occurs when it is dark, but lighted.
First, let's determine the total number of fatal crashes in normal weather:
- Total fatal crashes in normal weather = 14,307 (Daylight) + 5,875 (Dark, but lighted) + 8,151 (Dark) + 1,183 (Dawn/Dusk) + 65 (Other)
Next, we identify the number of these crashes that occurred when it is dark, but lighted:
- Fatal crashes in normal weather (dark, but lighted) = 5,875
The probability is then calculated as:
[tex]\[ \text{Probability} = \frac{\text{Number of fatal crashes in normal weather (dark, but lighted)}}{\text{Total number of fatal crashes in normal weather}} \][/tex]
Substituting in the values, we get:
[tex]\[ \text{Probability} = \frac{5,875}{29,581} \approx 0.199 \][/tex]
Therefore, the probability is approximately 0.199.
### Part (b)
Among fatal crashes when it is dark, but lighted, we want to know the probability that a randomly selected fatal crash occurs in normal weather.
First, determine the total number of fatal crashes when it is dark, but lighted across all weather conditions:
- Total fatal crashes (dark, but lighted) = 5,875 (Normal) + 497 (Rain) + 51 (Snow/Sleet) + 54 (Other) + 255 (Unknown)
Next, we use the number of dark, but lighted crashes that occur in normal weather (5,875) from part (a).
The probability is then calculated as:
[tex]\[ \text{Probability} = \frac{\text{Number of fatal crashes in normal weather (dark, but lighted)}}{\text{Total number of fatal crashes (dark, but lighted)}} \][/tex]
Substituting in the values, we get:
[tex]\[ \text{Probability} = \frac{5,875}{6,732} \approx 0.873 \][/tex]
Therefore, the probability is approximately 0.873.
### Part (c)
We need to determine whether it is more dangerous (higher probability of fatal crashes) in normal weather or in snow/sleet when it is dark (without light).
First, we find the probability of fatal crashes in normal weather when it is dark:
- Fatal crashes in normal weather (dark) = 8,151
- Total fatal crashes in normal weather = 29,581 (from part (a))
The probability is:
[tex]\[ \text{Probability (Dark in normal weather)} = \frac{8,151}{29,581} \approx 0.276 \][/tex]
Next, we find the probability of fatal crashes in snow/sleet when it is dark:
- Fatal crashes in snow/sleet (dark) = 156
- Total fatal crashes in snow/sleet = 445 (219 + 51 + 156 + 17 + 2)
The probability is:
[tex]\[ \text{Probability (Dark in snow/sleet)} = \frac{156}{445} \approx 0.351 \][/tex]
Comparing these probabilities, we see that the probability of a fatal crash is higher when it is dark in snow/sleet (0.351) than in normal weather (0.276).
Therefore, the correct answer is:
C. The dark is more dangerous in snow/sleet because the conditional probability of a fatal crash in snow/sleet is greater than that of a fatal crash in normal weather.
