Calculating the Probability of Exactly 50 Heads in 100 Coin Tosses: A Comprehensive Guide

Understanding the probability of achieving a specific outcome in a series of coin tosses is a fundamental concept in probability theory. When tossing 100 coins, what is the probability of landing exactly 50 heads? This question can be approached using the binomial probability formula or the normal approximation of the binomial distribution. Let's explore both methods in detail.

Using the Binomial Probability Formula

The binomial probability formula is a powerful tool for calculating the probability of a specific number of successes in a series of independent trials. In the case of tossing 100 coins and wanting exactly 50 heads, the formula is given by:

P(X 50) n choose k * p^k * (1 - p)^(n - k)

where:

n 100 (number of trials or coin tosses) k 50 (number of successes or heads) p 0.5 (probability of getting heads in each toss)

The binomial coefficient, denoted by n choose k or binom{n}{k}, represents the number of ways to choose k successes out of n trials. It is calculated as:

binom{n}{k} n! / (k!(n - k)!)

Step-by-Step Calculation

Calculate the binomial coefficient:

binom(100, 50) 100! / (50! * 50!)

Calculate the probability terms: P(h) 0.550 P(t) (1 - 0.5)50 0.550 Combine everything:

P(X 50) binom(100, 50) * 0.550 * 0.550 binom(100, 50) * 0.5100

Calculate binom(100, 50):

binom(100, 50) ≈ 1.008913 * 1029

Final Probability:

P(X 50) ≈ 1.008913 * 1029 * 0.5100

0.5100 1 / 2100 ≈ 7.888 * 10-31

P(X 50) ≈ 1.008913 * 1029 * 7.888 * 10-31 ≈ 0.079589

Normal Approximation of the Binomial Distribution

For large values of n, the binomial distribution can be approximated by the normal distribution. This approximation is particularly useful when dealing with large numbers of trials.

The mean (μ) and variance (σ2) of the binomial distribution are given by:

μ np 100 * 0.5 50 σ2 np(1 - p) 100 * 0.5 * 0.5 25

The standard deviation (σ) is:

σ √σ2 √25 5

To approximate the probability of getting exactly 50 heads, we use the continuity correction:

Pr(49.5 ≤ X ≤ 50.5) ≈ Pr((49.5 - μ) / σ ≤ Z ≤ (50.5 - μ) / σ)

Substituting the values:

Pr((49.5 - 50) / 5 ≤ Z ≤ (50.5 - 50) / 5)

Pr(-0.1 ≤ Z ≤ 0.1)

This can be further simplified as:

Pr(Z ≤ 0.1) - Pr(Z ≤ -0.1)

Using the standard normal distribution table, we find:

Pr(Z ≤ 0.1) ≈ 0.5398

Pr(Z ≤ -0.1) ≈ 0.4602

Pr(-0.1 ≤ Z ≤ 0.1) ≈ 0.5398 - 0.4602 0.0796

Verification Through Simulation

Using an Excel spreadsheet with a large number of trials (1 million), the empirical probability of getting exactly 50 heads in 100 coin tosses was found to be approximately 7.9%.

Therefore, the probability of getting exactly 50 heads when tossing 100 coins is approximately 0.0796 or 7.96%.

Related Keywords

binomial probability coin toss standard normal approximation