Final answer:
To estimate the probability that at least 30 people experience flu symptoms as an adverse reaction to the drug, we need to use the binomial distribution formula. We calculate the probability of X < 30 and subtract it from 1 to find the probability of X ≥ 30. This suggests the likelihood of at least 30 people experiencing flu symptoms as an adverse reaction to the drug.
Step-by-step explanation:
To estimate the probability that at least 30 people experience flu symptoms as an adverse reaction to the drug, we need to use the binomial distribution formula. The formula is given by:
P(X ≥ k) = 1 - P(X < k)
In this case, k = 30. So we need to find P(X < 30) and subtract it from 1 to get the desired probability. Let's calculate the values:
P(X ≥ 30) = 1 - P(X < 30)
P(X < 30) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 29)
We can use the binomial probability formula to calculate each individual term in the sum. The formula is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
C(n, k) = n! / (k! * (n - k)!)
n = total number of trials
k = number of successful trials
p = probability of success on a single trial
In this case, n = number of people treated, k = 0 to 29 (for each term in the sum), and p = probability of flu symptoms for a person not receiving any treatment. Let's calculate the individual terms and sum them up to get P(X < 30).
Finally, we can substitute the calculated value of P(X < 30) into the first formula to find P(X ≥ 30).
(a) P(X ≥ 30) = 1 - P(X < 30)
(b) The result from part (a) suggests the probability that at least 30 people experience flu symptoms as an adverse reaction to the drug.