We can solve these problems using the binomial distribution, since we are dealing with a fixed number of trials (210 users) and a binary outcome (experiencing or not experiencing a side effect).

Part 1: Probability of finding between 8 and 12 cases with side effects

Let:

• $n = 210$ (the number of trials or users),
• $p = 0.05$ (the probability of experiencing a serious side effect),
• $X$ be the random variable representing the number of cases with side effects.

We are asked to find the probability of $X$ being between 8 and 12 cases, inclusive. Mathematically, this is:

$P(8 \leq X \leq 12) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)$

The binomial probability mass function is given by:

$P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$

Part 2: Probability of finding more than 16 cases with side effects

We need to calculate:

$P(X > 16) = 1 - P(X \leq 16)$

Let's calculate both parts using Python.### Results:

1. The probability of finding between 8 and 12 cases with side effects is approximately 0.5744.
2. The probability of finding more than 16 cases with side effects is approximately 0.0356.

Would you like further details on the calculations, or do you have any other questions?

Related Questions:

1. What is the probability of finding exactly 10 cases with side effects?
2. What is the expected number of users experiencing side effects?
3. How does the probability change if the sample size increases to 300 users?
4. What is the probability of finding fewer than 5 cases with side effects?
5. How would the results differ if the probability of side effects increased to 10%?

Tip:

In binomial problems, for large $n$ and small $p$, the Poisson approximation can be a useful shortcut to avoid complex calculations.