Question

If you flip a 20-sided die 15 times, calculate the probability that 2 of the rolls land on the number 4. Show that normal distribution does not approximate this well by calculating what estimate we would get using a normal approximation.

Answer 1

μ = 15 * (1/20) = 0.75

σ = sqrt(15 * (1/20) * (19/20)) ≈ 0.98

Now, we can use the normal distribution to approximate the probability. We can convert the desired value of 2 rolls landing on the number 4 into a standard score (z-score) using the formula:

z = (x - μ) / σ

Where x is the value we want to approximate (2 in this case).

Calculating the z-score, we get:

z = (2 - 0.75) / 0.98 ≈ 1.28

Using a z-table or a calculator, we can find the approximate probability associated with a z-score of 1.28.

However, it's important to note that the normal approximation is not accurate for small sample sizes or when the success probability (p) is close to 0 or 1. In this case, the sample size is relatively small (15) and the success probability is low (1/20). So, the normal approximation may not provide an accurate estimate in this scenario.

Therefore, to get a more precise estimate, it's better to use the binomial probability formula.

Explanation: