Answer to Question #167885 in Statistics and Probability for Grace

Question

There are 20 machines in a large plant. The machines operate independently of one another, breakdowns occur at random. The probability of a breakdown for each machine in one day’s operation is 0.10. Find the probability that exactly 4 breakdowns occur in one day

Answer 1

By condition,

$p = 0.1 \Rightarrow q = 1 - p = 0.9$

Then, according to Bernoulli's formula, the wanted probability is

${P_{20}}\left( 4 \right) = C_{20}^4{p^4}{q^{20 - 4}} = \frac{{20!}}{{4!16!}} \cdot {0.1^4} \cdot {0.9^{16}} =\\= \frac{{17 \cdot 18 \cdot 19 \cdot 20}}{{24}} \cdot {0.1^4} \cdot {0.9^{16}} \approx 0.0898$

Answer: $\approx 0.0898.$