The table summarizes results from 985 pedestrian deaths that were caused by automobile accidents.

\begin{tabular}{|c||c|c|}
\hline
\begin{tabular}{c}
Driver \\
Intoxicated?
\end{tabular} & \begin{tabular}{c}
Pedestrian \\
Intoxicated?
\end{tabular} \\
\cline { 2 - 3 } & Yes & No \\
\hline Yes & 54 & 73 \\
\hline No & 233 & 625 \\
\hline
\end{tabular}

If one of the pedestrian deaths is randomly selected, find the probability that the pedestrian was intoxicated or the driver was not intoxicated. Please enter a decimal to 4 decimal places.

[tex]\[ \text{Probability} = \square \][/tex]



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]