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Chapter 8: Problem 22

If you roll a die 200 times, what is the approximate probability that you will roll more than 40 ones? (Round your answer to two decimal places.)

Short Answer

Expert verified
The approximate probability of rolling more than 40 ones in 200 rolls of a fair six-sided die is P(X > 40) = 1 - P(X 鈮� 40). To calculate P(X 鈮� 40), use the binomial probability formula and sum up the probabilities of rolling 40 or fewer ones: P(X 鈮� 40) = P(X = 0) + P(X = 1) + ... + P(X = 40), where n = 200, p = 1/6, and q = 5/6. Once you find P(X 鈮� 40), subtract it from 1 and round the result to two decimal places.

Step by step solution

01

Identify the binomial distribution parameters

We first identify the parameters that will be used in the binomial probability formula. In this case, we have: n = 200 (number of trials, i.e., the number of times the die is rolled) p = 1/6 (probability of success, i.e., probability of rolling a 1 in each trial) q = 5/6 (probability of failure, i.e., probability of not rolling a 1 in each trial) We want to find the probability of rolling more than 40 ones (more than 40 successes), i.e., P(X > 40). To find it, we will calculate the complementary probability, P(X 鈮� 40), and then subtract it from 1, so that P(X > 40) = 1 - P(X 鈮� 40).
02

Apply the binomial probability formula (using complement rule)

We can calculate P(X > 40) by finding the complement of P(X 鈮� 40). In other words, we will find the probability of rolling 40 or fewer ones and then subtract it from 1 to get the probability of rolling more than 40 ones. The binomial probability formula is: P(X = k) = \( \binom{n}{k} \) * p^k * q^(n-k) Using this formula, we will calculate P(X = k) for each k from 0 to 40, and then sum up these probabilities to get P(X 鈮� 40).
03

Calculate the probabilities for each k and sum them up

Using the formula from Step 2, we will calculate probabilities and sum them up: P(X 鈮� 40) = P(X = 0) + P(X = 1) + ... + P(X = 40) We can use a calculator or statistical software, such as Python or Excel, to perform these calculations and obtain P(X 鈮� 40).
04

Finding P(X > 40) and rounding to two decimal places

Once we have calculated P(X 鈮� 40), we find P(X > 40) by subtracting it from 1: P(X > 40) = 1 - P(X 鈮� 40) Finally, round the probability to two decimal places as instructed in the exercise. The approximate probability of rolling more than 40 ones in 200 rolls of a fair six-sided die will be obtained.

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Most popular questions from this chapter

Pastimes A survey of all the students in your school yields the following probability distribution, where \(X\) is the number of movies that a selected student has seen in the past week: \begin{tabular}{|r|c|c|c|c|c|} \hline Number of Movies & 0 & 1 & 2 & 3 & 4 \\ \hline Probability & \(.5\) & \(.1\) & \(.2\) & \(.1\) & \(.1\) \\ \hline \end{tabular} Compute the expected value \(\mu\) and the standard deviation \(\sigma\) of \(X\). (Round answers to two decimal places.) For what percentage of students is \(X\) within two standard deviations of \(\mu\) ?

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