Question

A die is rolled 360 times. Let say that you want to use normal approximation to find the probability that the number of 4 was rolled less than 100 times. You need to find the probability that X<100. Apply the continuity correction in this case.

Answer 1

To approximate the probability of rolling a 4 less than 100 times in 360 rolls of a die using the normal approximation, we first need to identify the parameters of the binomial distribution.

1. **Identify the Binomial Distribution:**
The random variable \(X\), representing the number of times a 4 is rolled, follows a binomial distribution with parameters:
- \(n = 360\) (the number of trials)
- \(p = \frac{1}{6}\) (the probability of rolling a 4 in a single trial)

2. **Calculate the Mean and Standard Deviation:**
The mean (\(\mu\)) and standard deviation (\(\sigma\)) of a binomial distribution are given by:
\[
\mu = n \cdot p = 360 \cdot \frac{1}{6} = 60
\]
\[
\sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{360 \cdot \frac{1}{6} \cdot \frac{5}{6}} = \sqrt{360 \cdot \frac{5}{36}} = \sqrt{50} \approx 7.07
\]

3. **Apply the Continuity Correction:**
To find \(P(X < 100)\), we apply continuity correction. This means we will calculate \(P(X < 100.5)\) in the normal approximation.

4. **Standardize the Normal Variable:**
We convert the variable to the standard normal variable \(Z\):
\[
Z = \frac{X - \mu}{\sigma}
\]
For \(X < 100.5\):
\[
Z = \frac{100.5 - 60}{7.07} \approx \frac{40.5}{7.07} \approx 5.73
\]

5. **Find the Probability:**
We look up the value for \(Z = 5.73\) in the standard normal distribution table. Since \(Z = 5.73\) is extremely high, the probability \(P(Z < 5.73)\) is practically 1.

**Conclusion:**
Thus, the probability that the number of times a 4 is rolled is less than 100 is approximately:
\[
P(X < 100) \approx 1
\]

This indicates that it is very likely (almost certain) that fewer than 100 fours will be rolled in 360 rolls of the die.