Assume that the spacecraft is subject to enough electron bombardment to cause an electric discharge every 5th day, and that the first discharge occurs on day 3. Then, assume that each electric discharge has a probability 0.125 to affect the onboard circuitry. If three, or more, discharges affecting the circuit occurs, this will lead to circuit malfunction.
a) How likely is it that the onboard circuit survives for at least 52 days?
I have interpreted this question as a binomial distribution problem with Bernoulli trias that have probability p = 0.125.
I calculate the number of 5-day periods in these 52 days: 52-3/5 = 9 The probability of having 0,1, or 2 discharges in the first 45 days (9*5) using Binomial distribution would be 0.981. The overall probibility of having 0,1, or 2 discharges affecting the circuit in the first 52 days is P(X <= 2 in first 45 days) * P(X <= 2 in remaining 7 days) = 0.964 * 0.981 = 0.946 - Is this answer correct?
b) Electron bombardment varies with solar activity. If solar activity is high, an electric discharge will occur every 3rd day. The probability of high solar activity is 0.2. If solar activity is low,an electric discharge will only occur every 10th day.
Low solar activity scenario: At most there will be 5 discharges. P[discharge affecting the circuit does not occur] = 0.875. P[survival for 52 or more] = P[no more than 2 discharges affecting the circuit in each 5-day period from day 3 to day 52 | low solar activity]
High solar activity scenario: Electric discharge occurs every 3rd day -> at most 17 discharges affecting the circuit. P[discharge affecting the circuit does not occur] = 1-0.125 = 0.875 P[survival for 52 days or more | high solar activity] = P[no more than 2 discharges affecting the circuit in each 5-day period from day 3 to day 52 | high solar activity]. I don't know how to continue to get the full answer.