Final answer:
The probability that exactly 3 out of 7 cars with a protection system will be recovered, given a 50% chance of recovery per car, is 0.2734 (option A).
Step-by-step explanation:
The question is asking for the probability that exactly 3 out of 7 cars with a protection system will be recovered, given that a car with a protection system will be recovered 50% of the time.
This is a typical binomial probability problem, where the number of trials is 7 (the number of cars), the number of successes sought is 3 (the cars recovered), and the probability of success on a single trial is 0.5 (50% chance of recovery).
To calculate the probability, we use the binomial probability formula:
P(X = k) = C(n, k) × p^k × (1-p)^(n-k),
where:
- P(X = k) is the probability of k successes in n trials,
- C(n, k) is the number of combinations of n items taken k at a time, often written as n choose k,
- p is the probability of success on a single trial,
- n is the total number of trials,
- k is the number of successful trials.
Plugging in the values we have:
P(X = 3) = C(7, 3) × 0.5^3 × (1-0.5)^(7-3),
Calculating this we get:
P(X = 3) = 35 × 0.125 × 0.0625 = 0.2734375,
So, the probability that exactly 3 out of 7 cars will be recovered is 0.2734, which corresponds to option A.