An Air Force study indicates that the probability of a disaster such as the January 28, 1986 explosion of the space shuttle Challenger was 1 in 35. The Challenger flight was the 25th mission.

a. How many disasters would you expect in the first 25 flights?
b. Use the normal approximation to estimate the probability of at least one disaster in 25 missions.

a. To calculate the expected number of disasters in the first 25 flights, you can use the probability of a disaster occurring (1/35) and multiply it by the number of flights (25).

Expected number of disasters = Probability of a disaster * Number of flights
Expected number of disasters = (1/35) * 25
Expected number of disasters = 0.714

Therefore, you would expect approximately 0.714 disasters in the first 25 flights.

b. To use the normal approximation to estimate the probability of at least one disaster in 25 missions, we need to find the complement of the probability of no disasters in 25 missions.

The probability of no disasters in 25 missions can be calculated using the formula for the probability of independent events occurring. In this case, the probability of no disasters is given by:

Probability of no disasters = (1 - Probability of a disaster) ^ Number of flights

Probability of no disasters = (1 - (1/35)) ^ 25

Using a calculator, you can evaluate this expression to approximately 0.694.

The probability of at least one disaster in 25 missions is equivalent to 1 minus the probability of no disasters:

Probability of at least one disaster = 1 - Probability of no disasters
Probability of at least one disaster = 1 - 0.694

Using a calculator, you can find that the probability of at least one disaster in 25 missions is approximately 0.306 or 30.6%.