The data in the table below summarize results from 154 pedestrian deaths that were caused by accidents. If two different deaths are randomly selected without replacement, find the probability that they both involved intoxicated drivers. Is such an event unlikely?
| Pedestrian Intoxicated? | Yes | No |
|---|---|---|
| Driver Intoxicated? Yes | 49 | 42 |
| Driver Intoxicated? No | 34 | 29 |
Answer (2)
The probability that both deaths involved intoxicated drivers is approximately 0.3472, which is not considered unlikely since it is greater than 0.05.
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To determine the probability that both randomly selected pedestrian deaths involved intoxicated drivers, we'll follow these steps:
Identify the total number of deaths involving intoxicated drivers : According to the data, there are 49 deaths where the driver was intoxicated and the pedestrian was also intoxicated. There are also 42 deaths with an intoxicated driver but a non-intoxicated pedestrian. Thus, the total number of deaths involving intoxicated drivers is: 49 + 42 = 91
Find the probability of selecting one death that involved an intoxicated driver : The total number of pedestrian deaths is 154. So, the probability of choosing one death that involved an intoxicated driver is: 154 91
Find the probability of selecting a second death that involved an intoxicated driver, without replacement : After selecting one death involving an intoxicated driver, there are now 90 deaths left involving intoxicated drivers out of 153 total remaining deaths. So, the probability for the second selection is: 153 90
Calculate the probability of both selections involving intoxicated drivers : Multiply the probabilities of the two independent events: 154 91 × 153 90 = 23562 8190
Simplify the probability : Simplify the fraction: = 786 273 ≈ 0.3472
Therefore, the probability that both deaths involved intoxicated drivers is approximately 0.3472.
Determine if the event is unlikely : In probability, an event is typically considered unlikely if it has a probability of less than 0.05. Since 0.3472 is greater than 0.05, this event is not considered unlikely.
