Answer :
Sure! Let's solve this problem step-by-step.
First, we need to define the given data from the table:
- The total number of pedestrian deaths: [tex]\(985\)[/tex]
- The number of cases where the pedestrian was intoxicated: [tex]\(54 + 233 = 287\)[/tex]
- The number of cases where the driver was not intoxicated: [tex]\(233 + 625 = 858\)[/tex]
Now, let's define the events we are interested in:
1. Let [tex]\(A\)[/tex] be the event that the pedestrian was intoxicated.
2. Let [tex]\(B\)[/tex] be the event that the driver was not intoxicated.
We want to find the probability of [tex]\(A \cup B\)[/tex], that is, the probability that the pedestrian was intoxicated or the driver was not intoxicated.
We use the principle of inclusion and exclusion to find this probability:
[tex]\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \][/tex]
Where:
- [tex]\(P(A)\)[/tex] is the probability that the pedestrian was intoxicated.
- [tex]\(P(B)\)[/tex] is the probability that the driver was not intoxicated.
- [tex]\(P(A \cap B)\)[/tex] is the probability that both the pedestrian was intoxicated and the driver was not intoxicated.
Next, we calculate these probabilities:
1. The probability that the pedestrian was intoxicated:
[tex]\[ P(A) = \frac{\text{Number of intoxicated pedestrians}}{\text{Total pedestrian deaths}} = \frac{287}{985} \][/tex]
2. The probability that the driver was not intoxicated:
[tex]\[ P(B) = \frac{\text{Number of cases where driver not intoxicated}}{\text{Total pedestrian deaths}} = \frac{858}{985} \][/tex]
3. The probability that both the pedestrian was intoxicated and the driver was not intoxicated:
[tex]\[ P(A \cap B) = \frac{\text{Number of cases where both pedestrian was intoxicated and driver was not intoxicated}}{\text{Total pedestrian deaths}} = \frac{233}{985} \][/tex]
Substitute these values into the formula for [tex]\(P(A \cup B)\)[/tex]:
[tex]\[ P(A \cup B) = \frac{287}{985} + \frac{858}{985} - \frac{233}{985} \][/tex]
Combine and simplify the fractions:
[tex]\[ P(A \cup B) = \frac{287 + 858 - 233}{985} = \frac{912}{985} \][/tex]
Convert to decimal and round to 4 decimal places:
[tex]\[ P(A \cup B) \approx 0.9259 \][/tex]
Thus, the probability that the pedestrian was intoxicated or the driver was not intoxicated is:
[tex]\[ \boxed{0.9259} \][/tex]
First, we need to define the given data from the table:
- The total number of pedestrian deaths: [tex]\(985\)[/tex]
- The number of cases where the pedestrian was intoxicated: [tex]\(54 + 233 = 287\)[/tex]
- The number of cases where the driver was not intoxicated: [tex]\(233 + 625 = 858\)[/tex]
Now, let's define the events we are interested in:
1. Let [tex]\(A\)[/tex] be the event that the pedestrian was intoxicated.
2. Let [tex]\(B\)[/tex] be the event that the driver was not intoxicated.
We want to find the probability of [tex]\(A \cup B\)[/tex], that is, the probability that the pedestrian was intoxicated or the driver was not intoxicated.
We use the principle of inclusion and exclusion to find this probability:
[tex]\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \][/tex]
Where:
- [tex]\(P(A)\)[/tex] is the probability that the pedestrian was intoxicated.
- [tex]\(P(B)\)[/tex] is the probability that the driver was not intoxicated.
- [tex]\(P(A \cap B)\)[/tex] is the probability that both the pedestrian was intoxicated and the driver was not intoxicated.
Next, we calculate these probabilities:
1. The probability that the pedestrian was intoxicated:
[tex]\[ P(A) = \frac{\text{Number of intoxicated pedestrians}}{\text{Total pedestrian deaths}} = \frac{287}{985} \][/tex]
2. The probability that the driver was not intoxicated:
[tex]\[ P(B) = \frac{\text{Number of cases where driver not intoxicated}}{\text{Total pedestrian deaths}} = \frac{858}{985} \][/tex]
3. The probability that both the pedestrian was intoxicated and the driver was not intoxicated:
[tex]\[ P(A \cap B) = \frac{\text{Number of cases where both pedestrian was intoxicated and driver was not intoxicated}}{\text{Total pedestrian deaths}} = \frac{233}{985} \][/tex]
Substitute these values into the formula for [tex]\(P(A \cup B)\)[/tex]:
[tex]\[ P(A \cup B) = \frac{287}{985} + \frac{858}{985} - \frac{233}{985} \][/tex]
Combine and simplify the fractions:
[tex]\[ P(A \cup B) = \frac{287 + 858 - 233}{985} = \frac{912}{985} \][/tex]
Convert to decimal and round to 4 decimal places:
[tex]\[ P(A \cup B) \approx 0.9259 \][/tex]
Thus, the probability that the pedestrian was intoxicated or the driver was not intoxicated is:
[tex]\[ \boxed{0.9259} \][/tex]
