To find the probability of more than two engine failures on a certain transcontinental flight, we need to calculate the probability of exactly 0, 1, and 2 engine failures and subtract that from 1.
The probability of a single engine failure is 0.004. Since engine failures are assumed to be independent, we can 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 getting exactly k engine failures
- n is the number of trials (in this case, the number of engines, which is 4)
- k is the number of successes (engine failures) we are interested in
- p is the probability of success (engine failure) on a single trial
For exactly 0 engine failures, we have:
P(x=0) = C(4, 0) * (0.004)^0 * (1-0.004)^(4-0)
C(4, 0) is the number of ways to choose 0 out of 4, which is 1.
P(x=0) = 1 * 1 * (0.996)^4
For exactly 1 engine failure, we have:
P(x=1) = C(4, 1) * (0.004)^1 * (1-0.004)^(4-1)
C(4, 1) is the number of ways to choose 1 out of 4, which is 4.
P(x=1) = 4 * 0.004 * 0.996^3
For exactly 2 engine failures, we have:
P(x=2) = C(4, 2) * (0.004)^2 * (1-0.004)^(4-2)
C(4, 2) is the number of ways to choose 2 out of 4, which is 6.
P(x=2) = 6 * (0.004)^2 * (0.996)^2
Now we can calculate the probability of more than two engine failures:
P(more than two engine failures) = 1 - P(x=0) - P(x=1) - P(x=2)
P(more than two engine failures) = 1 - [1 * 1 * (0.996)^4] - [4 * 0.004 * 0.996^3] - [6 * (0.004)^2 * (0.996)^2]
Calculating this expression will give us the probability of experiencing more than two engine failures on a certain transcontinental flight.