Problem 13 - Algebra and Trigonometry

Question

Rolling Dice Six dice are rolled. Find the probability that 2 of them show a four.

Step-by-Step Solution

Answer

The probability is approximately 0.0321.

Step 1: Understanding the Problem

We need to find the probability that exactly 2 out of 6 rolled dice show a four. This is a classic binomial probability problem where each die roll can be considered a Bernoulli trial with two outcomes: showing a four or not showing a four.

Step 2: Defining Parameters for Binomial Distribution

In this scenario, the number of trials \( n = 6 \) (since we roll the dice 6 times), and the probability of success (showing a four) in each trial is \( p = \frac{1}{6} \) because there is one four on a die out of six faces.

Step 3: Setting up the Binomial Probability Formula

The binomial probability formula is \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( \binom{n}{k} \) is the binomial coefficient. Here, we want \( k = 2 \) successes (fours).

Step 4: Calculating the Binomial Coefficient

The binomial coefficient \( \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \). This represents the number of ways to choose which 2 dice out of 6 will show a four.

Step 5: Calculating the Probability of 2 Fours

Now plug into the formula: \( P(X = 2) = 15 \left( \frac{1}{6} \right)^2 \left( \frac{5}{6} \right)^{4} \). Calculate \( \left( \frac{1}{6} \right)^2 = \frac{1}{36} \) and \( \left( \frac{5}{6} \right)^4 = \frac{625}{1296} \).

Step 6: Final Probability Calculation

Multiply everything together: \( P(X = 2) = 15 \times \frac{1}{36} \times \frac{625}{1296} = \frac{9375}{279936} \). Simplifying this fraction gives \( \frac{125}{3888} \), which is approximately 0.0321.

Key Concepts

Binomial Distribution

A binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials. Think of it as a model that helps us predict the outcome of repetitive actions where each action has only two possible outcomes: success or failure.

In our dice problem, rolling each die is one of these binary events. Either each die result shows a four (success), or it doesn't (failure). The binomial distribution provides a way to calculate the probabilities of getting a certain number of successes in a series of trials. Here, we are interested in the probability of rolling exactly two fours out of six dice rolls.

Key parameters of the binomial distribution include:

• Number of trials, denoted by \( n \). In our scenario, \( n = 6 \) since we roll the dice six times.
• Probability of success in a single trial, represented by \( p \). For a four, \( p = \frac{1}{6} \), as a die has six faces.

Utilizing these parameters, the binomial distribution allows us to compute the probability for the desired number of successes.

Bernoulli Trials

Bernoulli trials form the foundation of the binomial distribution. A Bernoulli trial is an experiment or process that results in one of two outcomes, such as success or failure. Each trial is an independent event, meaning the outcome of one trial does not influence another.

In our dice-rolling example, each roll can be considered a Bernoulli trial where:

• Success is rolling a four.
• Failure is any other result (not a four).

Since a die has six faces and only one of them is a four, the probability of success (rolling a four) in each trial is \( p = \frac{1}{6} \). Conversely, the probability of failure (not rolling a four) is \( 1 - p = \frac{5}{6} \).

Bernoulli trials are significant because they provide the simple building blocks for more complex probability models, like binomial distributions.

Binomial Coefficient

The binomial coefficient is a key mathematical concept within binomial distributions. It calculates the number of different ways in which a specific number of successes can occur within a set number of trials, without regard to the order of successes. It is denoted as \( \binom{n}{k} \), pronounced "n choose k."

In our problem, the binomial coefficient \( \binom{6}{2} \) helps to determine how many different ways two fours can occur when rolling six dice. The formula is:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

For this problem, calculate \( \binom{6}{2} = \frac{6!}{2!\cdot(6-2)!} = 15 \). This means there are 15 different combinations of obtaining exactly two fours from six rolls.

The binomial coefficient is essential as it provides the number of successful outcomes that we need to calculate the total probability using the binomial distribution formula.