Questions: Consider the probability that no more than 81 out of 104 computers will not crash in a day. Assume the probability that a given computer will not crash in a day is 33% Specify whether the normal curve can be used as an approximation to the binomial probability by verifying the necessary conditions.

Solution Steps

To determine if the normal approximation can be used for the binomial distribution, we first calculate the values of \( np \) and \( nq \):

\[ np = n \cdot p = 104 \cdot 0.33 = 34.32 \]

\[ nq = n \cdot q = 104 \cdot 0.67 = 69.68 \]

The conditions for using the normal approximation to the binomial distribution are:

1. \( np \geq 5 \)
2. \( nq \geq 5 \)

From our calculations:

• \( np = 34.32 \) which is greater than 5.
• \( nq = 69.68 \) which is also greater than 5.

Since both conditions are satisfied, we conclude that the normal approximation can be used.

Final Answer