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To find the probability that two randomly selected pedestrian deaths both involved intoxicated drivers, we need to follow a series of steps involving the given data about pedestrian deaths. Let's go through these steps carefully.
### Step 1: Understanding the Data
The table presents pedestrian deaths caused by automobile accidents, classified based on whether the driver was intoxicated and whether the pedestrian was intoxicated.
The data from the table are:
- Drivers Intoxicated, Pedestrian Intoxicated (Yes, Yes): 58
- Drivers Intoxicated, Pedestrian Not Intoxicated (Yes, No): 66
- Drivers Not Intoxicated, Pedestrian Intoxicated (No, Yes): 220
- Drivers Not Intoxicated, Pedestrian Not Intoxicated (No, No): 632
### Step 2: Calculate the Total Pedestrian Deaths
First, we need to find the total number of pedestrian deaths by summing up all the values given in the table:
[tex]\[ \text{Total Pedestrian Deaths} = 58 + 66 + 220 + 632 = 976 \][/tex]
### Step 3: Calculate the Number of Pedestrian Deaths Involving Intoxicated Drivers
Next, we need to determine the total number of pedestrian deaths where the drivers were intoxicated. This is the sum of pedestrian deaths for the cases where the driver was intoxicated:
[tex]\[ \text{Total Deaths with Intoxicated Drivers} = 58 + 66 = 124 \][/tex]
### Step 4: Calculate the Probability of Selecting One Pedestrian Death Involving an Intoxicated Driver
To find the probability that one randomly selected pedestrian death involved an intoxicated driver, we divide the number of pedestrian deaths involving intoxicated drivers by the total number of pedestrian deaths:
[tex]\[ P(\text{One Death Involves Intoxicated Driver}) = \frac{124}{976} \approx 0.1270 \][/tex]
### Step 5: Calculate the Probability of Selecting Two Pedestrian Deaths Both Involving Intoxicated Drivers
To find the probability that two randomly selected pedestrian deaths both involved intoxicated drivers, we use the multiplication rule for independent events:
1. The probability that the first death involves an intoxicated driver is:
[tex]\[ P(A) = \frac{124}{976} \][/tex]
2. If the first death involves an intoxicated driver, then the probability that the second death also involves an intoxicated driver (considering one less intoxicated driver and one less total death) is:
[tex]\[ P(B|A) = \frac{123}{975} \][/tex]
By multiplying these probabilities, we get:
[tex]\[ P(\text{Both Deaths Involve Intoxicated Drivers}) = \frac{124}{976} \times \frac{123}{975} \][/tex]
### Step 6: Calculating the Result
The calculations involving the above probabilities yield:
[tex]\[ P(\text{Both Deaths Involve Intoxicated Drivers}) \approx 0.0160 \][/tex]
### Step 7: Round to Four Decimal Places
Finally, we report the probability rounded to four decimal places, which is:
[tex]\[ P(\text{Both Deaths Involve Intoxicated Drivers}) = 0.0160 \][/tex]
Therefore, the probability that two different pedestrian deaths both involved intoxicated drivers is [tex]\( \boxed{0.0160} \)[/tex].
### Step 1: Understanding the Data
The table presents pedestrian deaths caused by automobile accidents, classified based on whether the driver was intoxicated and whether the pedestrian was intoxicated.
The data from the table are:
- Drivers Intoxicated, Pedestrian Intoxicated (Yes, Yes): 58
- Drivers Intoxicated, Pedestrian Not Intoxicated (Yes, No): 66
- Drivers Not Intoxicated, Pedestrian Intoxicated (No, Yes): 220
- Drivers Not Intoxicated, Pedestrian Not Intoxicated (No, No): 632
### Step 2: Calculate the Total Pedestrian Deaths
First, we need to find the total number of pedestrian deaths by summing up all the values given in the table:
[tex]\[ \text{Total Pedestrian Deaths} = 58 + 66 + 220 + 632 = 976 \][/tex]
### Step 3: Calculate the Number of Pedestrian Deaths Involving Intoxicated Drivers
Next, we need to determine the total number of pedestrian deaths where the drivers were intoxicated. This is the sum of pedestrian deaths for the cases where the driver was intoxicated:
[tex]\[ \text{Total Deaths with Intoxicated Drivers} = 58 + 66 = 124 \][/tex]
### Step 4: Calculate the Probability of Selecting One Pedestrian Death Involving an Intoxicated Driver
To find the probability that one randomly selected pedestrian death involved an intoxicated driver, we divide the number of pedestrian deaths involving intoxicated drivers by the total number of pedestrian deaths:
[tex]\[ P(\text{One Death Involves Intoxicated Driver}) = \frac{124}{976} \approx 0.1270 \][/tex]
### Step 5: Calculate the Probability of Selecting Two Pedestrian Deaths Both Involving Intoxicated Drivers
To find the probability that two randomly selected pedestrian deaths both involved intoxicated drivers, we use the multiplication rule for independent events:
1. The probability that the first death involves an intoxicated driver is:
[tex]\[ P(A) = \frac{124}{976} \][/tex]
2. If the first death involves an intoxicated driver, then the probability that the second death also involves an intoxicated driver (considering one less intoxicated driver and one less total death) is:
[tex]\[ P(B|A) = \frac{123}{975} \][/tex]
By multiplying these probabilities, we get:
[tex]\[ P(\text{Both Deaths Involve Intoxicated Drivers}) = \frac{124}{976} \times \frac{123}{975} \][/tex]
### Step 6: Calculating the Result
The calculations involving the above probabilities yield:
[tex]\[ P(\text{Both Deaths Involve Intoxicated Drivers}) \approx 0.0160 \][/tex]
### Step 7: Round to Four Decimal Places
Finally, we report the probability rounded to four decimal places, which is:
[tex]\[ P(\text{Both Deaths Involve Intoxicated Drivers}) = 0.0160 \][/tex]
Therefore, the probability that two different pedestrian deaths both involved intoxicated drivers is [tex]\( \boxed{0.0160} \)[/tex].
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