According to insurance records a car with a certain protection system will be recovered 89% of the time. Find the probability that 3 of 7 stolen cars will be recovered.
Use the binomial theorem. The probability that out of 7 stolen cars, the first three will be recovered is 0.89^3 x (1-0.89)^4 (because you also have to calculate the probability that the last four will NOT be recovered). That is 0.000103. But it's not just the first three that could have been recovered: you also have to work out in how many ways those three recovered ones could have been distributed among the seven, and multiply your answer by that. The number of ways it could have happened is (7!) / (3!)(4!)), where N! means N x (N-1) n (N-2) x ... 2 x 1. So the number of ways it could have happened is (7x6x5x4x3x2x1)/((4x3x2x1)x(3x2x1)), which is (7x6x5)/(3x2x1) = 35. Multiply that by the probability you worked out earlier, and you'll get 0.003613 - and that's your answer.
my evaluation worked out to be same answer. thought it was low until I was able to compare it with another source.
Statistics
Oh, stolen cars? That's not a laughing matter, but I'll do my best to bring some humor to your question!
To find the probability that 3 out of 7 stolen cars will be recovered, we can use the binomial probability formula. Let me calculate it for you:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- n is the total number of stolen cars (7 in this case)
- k is the number of stolen cars that will be recovered (3 in this case)
- p is the probability of a car being recovered (89% or 0.89 in decimal form)
Now let's calculate it using the formula!
P(X = 3) = (7 choose 3) * 0.89^3 * (1-0.89)^(7-3)
Calculating this, we get:
P(X = 3) ≈ 0.27576
So, the probability that 3 out of 7 stolen cars will be recovered is approximately 0.27576.
To find the probability that exactly 3 out of 7 stolen cars will be recovered, we can use the binomial probability formula. The formula is:
P(X = k) = C(n, k) * p^k * q^(n-k)
Where P(X = k) is the probability of getting exactly k successes, n is the total number of trials, p is the probability of success, q is the probability of failure, and C(n, k) is the number of possible combinations of k successes out of n trials.
In this case, n = 7 (total number of stolen cars), k = 3 (number of stolen cars recovered), p = 0.89 (probability that a stolen car with the protection system is recovered), and q = 1-p = 0.11 (probability that a stolen car with the protection system is not recovered).
Applying the formula:
P(X = 3) = C(7, 3) * 0.89^3 * 0.11^(7-3)
To calculate C(7, 3), we can use the combination formula:
C(n, k) = n! / (k!(n-k)!)
C(7,3) = 7! / (3!(7-3)!) = 7! / (3!4!) = (7 * 6 * 5) / (3 * 2 * 1) = 35
Substituting the values into the formula:
P(X = 3) = 35 * 0.89^3 * 0.11^4
Calculating the expression:
P(X = 3) = 35 * 0.704969 * 0.014641
P(X = 3) ≈ 0.0726
Therefore, the probability that exactly 3 out of 7 stolen cars will be recovered is approximately 0.0726.