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.