Let X be the number of people who experience mild side effects after taking the drug. X follows a binomial distribution with parameters n = 100 (the number of people taking the drug) and p = 0.75 (the probability of mild side effects).
To find the probability that more than 70 people will experience mild side effects, we need to calculate:
P(X > 70) = 1 - P(X ≤ 70)
Using the cumulative distribution function (CDF) of the binomial distribution, we can find:
P(X ≤ 70) = F(70) = ∑(i=0 to 70) [nCi * p^i * (1-p)^(n-i)]
where nCi is the binomial coefficient for choosing i items out of n.
We can use a software or calculator to calculate this sum, or use an approximation like the normal approximation to the binomial distribution.
Assuming normal approximation is appropriate, with mean μ = np = 75 and standard deviation σ = sqrt(np(1-p)) = 3.354, we can calculate the z-score:
z = (70.5 - μ) / σ = (70.5 - 75) / 3.354 = -1.33
Using a standard normal distribution table or a calculator with normal distribution functions, we can find the probability of z-score less than -1.33 is 0.0918, which is the probability of 70 or fewer people experiencing mild side effects.
Therefore, the probability of more than 70 people experiencing mild side effects is:
P(X > 70) = 1 - P(X ≤ 70) = 1 - 0.0918 = 0.9082
Using the binomial distribution, there is a probability of approximately 0.9082 that more than 70 people will experience mild side effects out of 100 people who take the drug.