Probability of Winning at Least 8 Times in 20 Games: A Comprehensive Guide

Imagine a scenario where you are playing a game with a winning probability of only 0.3. If you play this game 20 times, what is the probability that you will win at least 8 times? Let's explore this using the principles of binomial probability and distribution.

Understanding Binomial Distribution

In this scenario, we are dealing with a binomial distribution, which is a probability distribution of the number of successes in a fixed number of independent trials, where each trial has only two outcomes – a success or a failure. Here, each game is an independent trial, and the probability of success (winning the game) is 0.3.

Binomial Probability Formula

The probability of winning exactly k times out of n trials is given by the formula:

P(X k) {nchoose k} p^k (1-p)^{n-k}

Where:

n 20 (the number of trials, i.e., games played) p 0.3 (the probability of winning a single game) {nchoose k} frac{n!}{k!(n-k)!} (the binomial coefficient)

Calculating the Probability

Instead of calculating the probabilities of winning exactly 8, 9, …, 20 times separately, we can find the probability of winning at least 8 times by subtracting the probability of winning 7 times or fewer from 1. This can be mathematically expressed as:

P(X geq 8) 1 - P(X leq 7)

The probability of winning 7 times or fewer can be calculated using the formula for binomial probability, summed up from 0 to 7.

Step-by-Step Calculation

Calculate the probability of winning 0 through 7 times:

P(X 0) {20choose 0} 0.3^0 0.7^{20} P(X 1) {20choose 1} 0.3^1 0.7^{19} P(X 2) {20choose 2} 0.3^2 0.7^{18} P(X 3) {20choose 3} 0.3^3 0.7^{17} P(X 4) {20choose 4} 0.3^4 0.7^{16} P(X 5) {20choose 5} 0.3^5 0.7^{15} P(X 6) {20choose 6} 0.3^6 0.7^{14} P(X 7) {20choose 7} 0.3^7 0.7^{13}

Add these probabilities to find P(X leq 7):

P(X leq 7) sum_{k0}^{7} P(X k)

Calculate the probability of winning at least 8 times:

P(X geq 8) 1 - P(X leq 7)

Python Code Implementation

Below is a Python code snippet that implements the calculations:

import math n 20 p 0.3 q 1 - p def binomial_probability(k, p): return (n, k) * (p**k) * (q**(n - k)) P_X_leq_7 sum(binomial_probability(k, p) for k in range(8)) P_X_geq_8 1 - P_X_leq_7 print(P_X_geq_8)

Running this code will give you the probability of winning at least 8 times in 20 games, which, in this case, is approximately 0.073 or 7.3%.

Conclusion

The probability of winning at least 8 times in 20 games with a 0.3 winning probability can be calculated using binomial distribution principles. By following the step-by-step methodology, you can determine the exact probability and understand the underlying logic better. This approach not only provides a clear mathematical solution but also helps in applying similar techniques to other similar scenarios.