According to the National Transportation Safety Board, 10% of all major automobile crashes result in serious injury to at least one person involved in the crash. The Georgia Department of Transportation reports that there are approximately 20 major automobile crashes per month in the Metro Atlanta area.
c. What is the probability that, in 20 major crashes, no serious injuries occur?
d. What is the probability that, in 20 major crashes, at least one serious injury occurs?
To find the probability of an event, you need to know the total number of possible outcomes and the number of favorable outcomes. In this case, the probability of serious injury occurring in a major crash is given as 10%.
c. To find the probability that no serious injuries occur in 20 major crashes, we need to find the probability of the opposite event (at least one serious injury). The opposite event is easier to calculate because it is the complement of the event we want to find. Hence,
Probability (no serious injuries) = 1 - Probability (at least one serious injury)
To find the probability of at least one serious injury, we can use the complement rule.
Probability (at least one serious injury) = 1 - Probability (no serious injury)
Since the probability of serious injury in a major crash is given as 10% or 0.10, the probability of no serious injury in a major crash is 1 - 0.10 = 0.90.
Now, we can find the probability of no serious injuries in 20 major crashes using the formula for independent events:
Probability (no serious injuries in 20 major crashes) = (Probability (no serious injury)) ^ (Number of major crashes)
Probability (no serious injuries in 20 major crashes) = (0.90) ^ 20
Calculating this gives us the answer.
d. To find the probability that at least one serious injury occurs in 20 major crashes, we can use the formula for the complement event as mentioned earlier:
Probability (at least one serious injury in 20 major crashes) = 1 - Probability (no serious injuries)
We have already calculated the probability of no serious injuries in 20 major crashes in part c. We can use that value to find the probability of at least one serious injury by subtracting it from 1.
Probability (at least one serious injury in 20 major crashes) = 1 - Probability (no serious injuries)
Calculating this gives us the answer.