Final answer:
To calculate the probability using the Poisson distribution, we can use the probabilities given for earthquakes of magnitudes at least 6.0 and in the interval (4.9,6). The changes in assumptions about quake magnitudes would affect the probabilities of X and Y, ultimately affecting the probability of both events happening.
Step-by-step explanation:
Let's use the Poisson distribution to calculate the probability.
Part (a)
Let's define the random variable X as the number of earthquakes of magnitude at least 6.0 within 365 days, and Y as the number of earthquakes with magnitude in the interval (4.9,6) within 365 days.
Given that the probability of a quake with a magnitude of at least 6.0 in a day is 0.01, and the probability of a quake with a magnitude in the interval (4.9,6) in a day is 0.02, we can calculate the probabilities of X and Y using the Poisson distribution.
The probability of one quake of magnitude at least 6.0 within 365 days is P(X=1) = (365 * 0.01) * e^(-365 * 0.01).
The probability of at least one quake with magnitude in the interval (4.9,6) within 365 days is P(Y≥1) = 1 - P(Y=0) = 1 - e^(-365 * 0.02).
The probability of both events happening is P(X=1 and Y≥1) = P(X=1) * P(Y≥1).
Part (b)
The changes in the assumptions about quake magnitudes would affect the probability by changing the probabilities of X and Y. If the probability of a quake with a magnitude of at least 6.0 increases, the probability of X occurring will increase. Similarly, if the probability of a quake with magnitude in the interval (4.9,6) increases, the probability of Y occurring will increase. This would ultimately affect the probability of both events happening.