Final answer:
To estimate the probability of the given events, we need to know the probability of the medicine being effective on a single patient and how to calculate the probability of multiple independent events. For event 6, the probability is 0.512. For event 7, the probability is 0.104.
Step-by-step explanation:
To estimate the probability of events 6 and 7, we need to understand the probability of the medicine being effective on a single patient and how to calculate the probability of multiple independent events. Since the medicine is effective for 80% of patients, the probability of it being effective for a single patient is 0.8. Now, let's calculate the probabilities for event 6 and event 7:
6. The probability of the medicine being effective on each of the three patients is calculated by multiplying the probabilities together because the events are independent. So, the probability is 0.8 multiplied by 0.8 multiplied by 0.8, which equals 0.512.
7. The probability of the medicine being effective on fewer than two of the next three patients can be calculated by finding the probability of zero or one patient being effective. We can use the binomial probability formula: P(X=k) = nCk * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success, and (1-p) is the probability of failure. In this case, n=3, k=0 or 1, and p=0.8. Plugging in the values, we get P(X=0) + P(X=1) = (3C0 * 0.8^0 * 0.2^3) + (3C1 * 0.8^1 * 0.2^2) = 0.008 + 0.096 = 0.104.