On January 28,1986, the space shuttle Challenger was totally enveloped in an explosive burn that destroyed the shuttle and resulted in the deaths of all seven astronauts aboard. A presidential commission assigned to investigate the accident concluded that the explosion was caused by the failure of the O-ring seal in the joint between the two lower segments of the right solid-fuel rocket booster. In a report madc one year prior to the catastrophe, the National Aeronautics and Space Administration (NASA) claimed that the probability of such a failure was about '/ji,,,,,,,,), or about once in every 60,000 flights. But a risk assessment study conducted for the Air Force at about the same time assessed the probability of shuttle catastrophe due to booster rocket "burn-through" to be '/,,, or about once in every 35 missions.

a. The catastrophe occurred on the twenty-fifth shuttle mission. Assuming NASA's failure-rate estimate was accurate, compute the probability that no disasters would have occurred during 25 shuttle missions.

b. Repeat part

a, but use the Air Force's failure-rate estimate.

C. What conditions must exist for the probabilities, parts a and

b, to be valid?

d. Given the events of January 28, 1986, which risk assessment-NASA's or the Air Force's-appears to be more appropriate? [Hint: Consider the complement of the events, parts a and b.]

e. After making improvcments in the shuttle's systems over the late 1980s and early 1990s, NASA issued a report in 1993 in which the risk of catastrophic failure of the shuttle's main engine was assessed for each mission at 1 in 120. ("Laying Odds on Shuttle Disaster," Chance, Fall 1993.) Use this risk assessment and the binomial probability distribution to find the probability of at least one catastrophic failure in the next 10 missions.