The Probability of Rolling the Number 3 at Least 5 Times in 10 Dice Rolls: A Comprehensive Guide

Understanding the probabilities in dice rolling can be a foundational skill in numerous fields, from statistics to game theory. This article explores the detailed calculation of the probability of rolling the number 3 at least 5 times when rolling a regular six-sided die 10 times. We will delve into the binomial distribution, the formula for probability, and the step-by-step process to arrive at the solution.

Introduction to Binomial Distribution

Before diving into the probability calculation, it's essential to understand the concept of a binomial distribution. A binomial distribution is a probability distribution of the number of successes in a fixed number of independent Bernoulli trials with the same probability of success.

Parameters and Variables

The parameters of our binomial distribution are as follows:

n: The number of trials. Here, n 10 (10 dice rolls). k: The number of successes. Here, k is at least 5 (rolling a 3 at least 5 times). p: The probability of success on a single trial. For rolling a 3, p 1/6.

Binomial Probability Formula

The probability of getting exactly k successes in n trials is given by:

P(X k) C(n, k) * p^k * (1-p)^(n-k)

Where C(n, k) is the binomial coefficient, calculated as:

C(n, k) n! / [k!(n-k)!]

Calculating the Probability of at Least 5 Successes

To find the probability of rolling the number 3 at least 5 times, we need to calculate the sum of probabilities for k ranging from 5 to 10:

P(X ≥ 5) P(X 5) P(X 6) P(X 7) P(X 8) P(X 9) P(X 10)

Step-by-Step Probability Calculation

Let's calculate the probability of each individual value of k starting from k 5.

P(X 5) C(10, 5) * (1/6)^5 * (5/6)^5 P(X 6) C(10, 6) * (1/6)^6 * (5/6)^4 P(X 7) C(10, 7) * (1/6)^7 * (5/6)^3 P(X 8) C(10, 8) * (1/6)^8 * (5/6)^2 P(X 9) C(10, 9) * (1/6)^9 * (5/6)^1 P(X 10) C(10, 10) * (1/6)^10 * (5/6)^0

Binomial Coefficients

The binomial coefficients for each value of k are:

C(10, 5) 252 C(10, 6) 210 C(10, 7) 120 C(10, 8) 45 C(10, 9) 10 C(10, 10) 1

Calculating Each Term

Now, let's calculate each probability step-by-step:

P(X 5) 252 * (1/6)^5 * (5/6)^5 ≈ 0.2001 P(X 6) 210 * (1/6)^6 * (5/6)^4 ≈ 0.0461 P(X 7) 120 * (1/6)^7 * (5/6)^3 ≈ 0.0077 P(X 8) 45 * (1/6)^8 * (5/6)^2 ≈ 0.0009 P(X 9) 10 * (1/6)^9 * (5/6)^1 ≈ 0.00005 P(X 10) 1 * (1/6)^10 * (5/6)^0 ≈ 0.000002

Summing Up the Probabilities

Finally, we sum these probabilities to get the overall probability:

P(X ≥ 5) ≈ 0.2001 0.0461 0.0077 0.0009 0.00005 0.000002 ≈ 0.2549

Conclusion

The probability of rolling the number 3 at least 5 times in 10 rolls of a die is approximately 0.2549, or 25.49%. This result demonstrates the power of the binomial distribution in calculating probabilities.