Question

Consider the probability that exactly 95 out of 158 computers will not crash in a day. Assume the probability that a given computer will not crash in a day is 56%. Approximate the probability using the normal distribution.

Answer 1

Solution:

Given that,

P = 0.56

1 - P = 0.44

n = 158

Here, BIN ( n , P ) that is , BIN (158, 0.564)

then,

n*p = 158 * 0.56 = 88.48 > 5

n(1- P) = 158 * 0.44 = 69.52 > 5

According to normal approximation binomial,

X Normal

Mean = = n*P = 158 * 0.56 = 88.48

Standard deviation = =n*p*(1-p) = 1581 * 0.56 * 0.44 = 6.2395

We using countinuity correction factor

P(X = a) = P( a - 0.5 < X < a + 0.5)

P(94.5 < x < 95.5) = P((94.5 - 88.48)/6.2395 ) < (x - ) /  < (95.5 - 88.48) /6.2395) )

= P(0.9648 < z < 1.1251)

= P(z <1.1251 - P(z < 0.9648)

= 0.8697 - 0.8327 using z - table,

= 0.03704

Probability = 0.03704