This problem is essentially a probability problem, where the probability of a serious side effect is known to be 5% (or $0.05$), and a sample of 230 users is being examined.

Let's break down the information provided:

• Probability of a serious side effect (p) = 5% = 0.05
• Sample size (n) = 230

Key Questions to Address:

1. Expected Number of Users with Side Effects: The expected number of users in the sample who experience a serious side effect can be found using the formula for expectation in a binomial distribution: $E(X) = n \times p$ Substituting the values: $E(X) = 230 \times 0.05 = 11.5$ So, the expected number of users experiencing a serious side effect is 11.5.

2. Variance and Standard Deviation: The variance of a binomial distribution is given by: $\text{Var}(X) = n \times p \times (1 - p)$ Substituting the values: $\text{Var}(X) = 230 \times 0.05 \times 0.95 = 10.925$ The standard deviation is the square root of the variance: $\text{Std Dev}(X) = \sqrt{10.925} \approx 3.305$

3. Probability of a Specific Number of Users with Side Effects: If you want to calculate the probability of a specific number of users experiencing a serious side effect, you would use the binomial probability formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ where $k$ is the number of users with side effects.

   If you have a specific $k$ in mind, we can calculate it.

Questions for Further Exploration:

1. What is the probability that exactly 10 users experience a serious side effect?
2. What is the probability that fewer than 8 users experience a serious side effect?
3. What is the probability that more than 15 users experience a serious side effect?
4. What is the probability that between 8 and 12 users experience a serious side effect?
5. How does the expected number of users with side effects change if the probability of side effects increases to 8%?

Tip:

For large sample sizes and small probabilities, the binomial distribution can often be approximated by a normal distribution for easier calculations.