Problem 23 - A First Course in Probability

Question

One thousand independent rolls of a fair die will be made. Compute an approximation to the probability that the number 6 will appear between 150 and 200 times inclusively. If the number 6 appears exactly 200 times, find the probability that the number 5 will appear less than 150 times.

Step-by-Step Solution

Answer

The probability of rolling a 6 between 150 and 200 times inclusively in 1000 independent rolls of a fair die is approximately 0.873. The conditional probability of rolling a 5 less than 150 times given that the number 6 appears exactly 200 times is approximately 0.181.

Step 1: Identify the binomial distribution parameters

For a fair die, the probability of rolling a specific number, in this case 6, is 1/6, and the probability of not rolling that number is 5/6. The number of independent trials is 1000. The binomial distribution has two parameters: n (the number of trials) and p (the probability of success or rolling a 6 in this case). n = 1000 p = 1/6 q = 5/6

Step 2: Calculate the mean and standard deviation for the normal approximation

Since we are dealing with a large sample size, we can use the normal approximation to the binomial distribution. The mean (μ) and standard deviation (σ) of a binomial distribution are given by: μ = n * p σ = sqrt(n * p * q) where q is the probability of not rolling a 6, which is 5/6: μ = 1000 * (1/6) = 166.67 σ = sqrt(1000 * (1/6) * (5/6)) = 11.55

Step 3: Calculate the probability of rolling a 6 between 150 and 200 times

Using the normal approximation, we will find the z-scores for both 150 and 200 occurrences of rolling a 6: z1 = (150 - μ) / σ = (150 - 166.67) / 11.55 = -1.44 z2 = (200 - μ) / σ = (200 - 166.67) / 11.55 = 2.89 Now, we will use a standard normal distribution table or calculator to find the probabilities associated with these z-scores, and then find the probability of rolling a 6 between 150 and 200 times: P(150 ≤ X ≤ 200) = P(-1.44 ≤ Z ≤ 2.89) ≈ 0.925 (-1.44) + 0.998 (2.89) = 0.873 So, the probability of rolling a 6 between 150 and 200 times inclusively is approximately 0.873.

Step 4: Calculate the conditional probability of rolling a 5 less than 150 times given the number 6 appears exactly 200 times

Given that the number 6 appears exactly 200 times, there are remaining 800 independent rolls for the other five faces with equal probability of 1/5 for each. Therefore, the new parameters are: n = 800 p = 1/5 q = 4/5 Calculate the mean and standard deviation for these new parameters: μ = n * p = 800 * (1/5) = 160 σ = sqrt(n * p * q) = sqrt(800 * (1/5) * (4/5)) = 10.95 Now, we calculate the z-score for rolling a 5 less than 150 times: z = (150 - μ) / σ = (150 - 160) / 10.95 = -0.91 Using a standard normal distribution table or calculator to find the probability associated with this z-score: P(X < 150 | X6 = 200) = P(Z < -0.91) ≈ 0.181 So, the probability of rolling a 5 less than 150 times given that the number 6 appears exactly 200 times is approximately 0.181.