# A hip joint replacement part is being stress-tested in a laboratory. The probability of successfully completing the test is 0.883. 11 randomly and independently chosen parts are tested. What is the probability that exactly two of the 11 parts successfully complete the test?

Question
Probability
A hip joint replacement part is being stress-tested in a laboratory. The probability of successfully completing the test is 0.883. 11 randomly and independently chosen parts are tested. What is the probability that exactly two of the 11 parts successfully complete the test?

2021-03-08
A hip joint replacement part is being stress-tested in a laboratory. Let us define
p = probability of successfully completing the test = 0.883
q = 1 − p = 0.117 ​
We want to find the probability that exactly two of the 11 parts successfully complete the test. A random variable X is said to follow binomial distribution if it assumes only non-negative values and its probability mass function is given by: ​
$$P(X = x) = p(x) = \bigg(\frac{n}{x}\bigg)p^{x}q^{n-x}, x = 0, 1, 2..., n$$
Here we choose 11 randomly and independently parts. Therefore, n = 11. ​
Now using the binomial probability we have the probability that exactly two of the 11 parts successfully complete the test is ​
$$P(X = 2) = \bigg(\frac{11}{2}\bigg)(0.883)^{2}(0.117)^{11-2} = 5.5 * (0.883)^{2} * (0.117)^{9} = 0.000000176$$
Result: P(X = 2) = 0.000000176

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