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Magazine Chapter 8: Problem 42
A die is rolled 12 times. Approximate the probability of rolling the following. Exactly 6 ones
Short Answer
Expert verified
P(X = 6) \approx 0.0543.
Step by step solution
01
Identify the Problem Type
This problem involves rolling a die multiple times and counting occurrences of a specific outcome, which is a scenario suitable for the Binomial probability distribution. We have a fixed number of trials (12 rolls), two possible outcomes on each trial (rolling a one, or not), and the probability of success (rolling a one) remaining the same across trials.
02
Define the Variables
Let X be the random variable representing the number of ones rolled. We will use the Binomial distribution with parameters: - n = 12 (number of independent trials)- p = \(\frac{1}{6}\) (probability of rolling a one in a single roll).We want to find P(X = 6), the probability of rolling exactly 6 ones.
03
Set Up the Binomial Probability Formula
The probability of getting exactly k successes in n trials in a Binomial distribution is given by the formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]In this problem, k = 6, n = 12, and p = \(\frac{1}{6}\).
04
Calculate the Binomial Coefficient
We calculate the binomial coefficient \(\binom{12}{6}\): \[ \binom{12}{6} = \frac{12!}{6!(12-6)!} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 924 \].
05
Compute the Probability
With the binomial coefficient, calculate P(X = 6):\[ P(X = 6) = \binom{12}{6} \left(\frac{1}{6}\right)^6 \left(\frac{5}{6}\right)^{6} \]\[ P(X = 6) = 924 \times \left(\frac{1}{6}\right)^6 \times \left(\frac{5}{6}\right)^6 \].Calculate these powers:- \(\left(\frac{1}{6}\right)^6 \approx 2.143 \times 10^{-5}\)- \(\left(\frac{5}{6}\right)^6 \approx 0.334898\).Use these to find P(X = 6):\[ P(X = 6) = 924 \times 2.143 \times 10^{-5} \times 0.334898 \ = 0.0543 \].
06
Conclusion
The probability of rolling exactly six ones in twelve rolls of a die is approximately 0.0543.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Probability
Probability is a measure that reflects the likelihood of a particular event happening. In the context of rolling a die, we are often concerned with events like rolling a specific number on the die. To calculate probability, we observe these key points:
- The probability of a single outcome (e.g., rolling a one) is determined by dividing the number of favorable outcomes by the total number of possible outcomes.
- For a fair six-sided die, the probability of rolling a specific number, like a one, is \(\frac{1}{6}\).
- When we want to find the probability of several occurrences of this outcome (like rolling a one multiple times), we use probability distributions.
Random Variable
A random variable is a way to quantify uncertain events. It assigns numerical values to each outcome in a probability experiment. In our scenario of rolling a die, a random variable helps model the situation mathematicaly.Key points about random variables include:
- They are functions mapping outcomes of a random process to real numbers.
- For this exercise, our random variable \(X\) represents the number of times we roll a one in 12 rolls of the die.
- Random variables can be classified as either discrete or continuous. Here, since we're counting the number of ones, we deal with a discrete random variable.
Binomial Coefficient
The binomial coefficient \(\binom{n}{k}\) represents the number of ways to choose \(k\) successes in \(n\) trials. It's an essential piece when working with binomial distributions.Let's explore some crucial aspects:
- The formula is: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
- In our exercise, \(n = 12\) and \(k = 6\), meaning we're interested in rolling a one exactly six times out of twelve.
- The calculation involves factorials, which are products of all positive integers up to a specified number. For example, \(12! = 12 \times 11 \times ... \times 1\).
