Problem 35 - Discrete Mathematics with Applications

Question

A die is rolled four times. Find the probability of obtaining: Exactly three sixes.

Step-by-Step Solution

Answer

The probability of obtaining exactly three sixes when a die is rolled four times is \(\frac{5}{54}\).

Step 1: Calculate the number of combinations

In this step, we will calculate C(n, x), which is the number of combinations of rolling the die four times and getting exactly three sixes: \(C(n, x) = \frac{n!}{x!(n-x)!} = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!}\) We know that, \(4! = 4 \times 3 \times 2 \times 1 = 24\) \(3! = 3 \times 2 \times 1 = 6\) \(1! = 1\) So, using these factorials: \(C(n, x) = \frac{24}{6 \times 1} = 4\)

Step 2: Calculate the probability of obtaining exactly three sixes

Now that we have the number of combinations, we can plug our values into the binomial probability formula: \(P(x) = C(n, x) * p^x * (1-p)^{(n-x)}\) Plugging in our values, we get: \(P(3) = 4 * (\frac{1}{6})^3 * (\frac{5}{6})^{(4-3)}\)

Step 3: Simplify and solve

Now, we just need to simplify and solve the equation to find the probability: \(P(3) = 4 * (\frac{1}{6})^3 * (\frac{5}{6})^1\) \(P(3) = 4 * \frac{1}{216} * \frac{5}{6}\) \(P(3) = \frac{20}{216}\) We can simplify the fraction by dividing both the numerator and denominator by the greatest common divisor (4): \(P(3) = \frac{5}{54}\) So, the probability of obtaining exactly three sixes when a die is rolled four times is \(\frac{5}{54}\).

Key Concepts

Combinatorics

Combinatorics is like a fancy word for counting different ways things can happen. Imagine you have a small basket with blocks of different colors. If you want to know how many unique ways you can pick a certain number of blocks, ''combinatorics'' helps you figure that out.

In the exercise, when a die is rolled four times, we're interested in how many ways we can get exactly three sixes. This is where ''combinations'' come in. Combinations are a way to count choices without worrying about the order. This is calculated using a special formula:

• The formula is: \( C(n, x) = \frac{n!}{x!(n-x)!} \)
• Here, ''n'' is the total rolls (4 in this case), and ''x'' is the number of times we want a six (3).
• The exclamation mark "!" is called a ''factorial'' and it means to multiply all the numbers from 1 up to that number.

So, in short, combinatorics gives us the tools to figure out the number of different ways we can achieve a specific result, like rolling three sixes.

Binomial Distribution

The ''binomial distribution'' is like a game plan for predicting how often a yes/no event will happen. It deals with scenarios where you have two outcomes: success and failure.

In our rolling dice exercise, getting a six can be considered as a success (yes), and not getting a six is a failure (no). For rolling a die four times, we use the binomial formula to calculate the probability of getting exactly three sixes:

• The main formula is: \( P(x) = C(n, x) \cdot p^x \cdot (1-p)^{n-x} \)
• \( C(n, x) \) is our combinations from the first part.
• ''p'' is the probability of success on one roll (rolling a six, which is \( \frac{1}{6} \)).
• The probability of failure (not rolling a six) is \( \frac{5}{6} \).

This distribution allows us to see the likelihood or chance of a given number of successes (like three sixes) in several trials.

Factorials

Factorials are like a shortcut for multiplying a series of descending whole numbers down to 1. This operation is super handy in calculating combinations and permutations, which are crucial in probability theory.

When you see a number with an exclamation mark, like 4! or 3!, it is saying to multiply that number by each whole number below it. Here's how it works:

• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)
• The concept of factorial is crucial for combinations in determining \( C(n, x) \). We used it to find how many different ways a specific outcome could happen when rolling a dice.

Factorials make mathematical calculations more efficient, especially when dealing with large numbers, allowing us to easily determine probabilities and outcomes in various settings.