Understanding Binomial Distribution: Expectation and Variance in a Fair Coin Toss

In probability theory, the binomial distribution is a fundamental concept used to model the number of successes in a fixed number of independent Bernoulli trials. One common example of this is the tossing of a fair coin. This article explores the binomial distribution in the context of a fair coin tossed 6 times, calculating the expectation (mean) and variance of the number of heads obtained.

Introduction to Binomial Distribution

The binomial distribution is characterized by two parameters: n (the number of trials) and p (the probability of success on each trial). In the context of a fair coin toss, each trial has two possible outcomes: heads (success) or tails (failure). For a fair coin, the probability of obtaining heads in a single toss is p 0.5.

Calculating Expectation (Mean)

The expectation (mean) of a binomial random variable X representing the number of heads in n coin tosses is given by the formula:

EX np

In the case of a fair coin tossed 6 times, we have:

n 6, p 0.5

Substituting these values into the formula, we get:

EX 6 × 0.5 3

Calculating Variance

The variance of a binomial random variable X is given by the formula:

VX npq

Here, q is the probability of failure, which is 1 - p. For a fair coin, q 0.5. Substituting the values, we get:

VX 6 × 0.5 × 0.5 1.5

Verification Using First Principles

To verify these calculations, we can list all possible outcomes for a coin tossed 6 times and calculate the expectation and variance step-by-step.

Step 1: List All Possible Outcomes

The total number of outcomes for 6 tosses of a fair coin is:

26 64

Here are the outcomes sorted by the number of heads:

No. of Heads Frequency 0 1 1 6 2 15 3 20 4 15 5 6 6 1

Step 2: Calculate Expectation (Mean)

The expectation (mean) EX is calculated as the sum of the product of the number of heads and their corresponding frequencies:

EX 0 × 1 1 × 6 2 × 15 3 × 20 4 × 15 5 × 6 6 × 1 80 / 32 2.5

Step 3: Calculate Variance

The variance VX is calculated using the formula:

VX EX2 - [EX]2

First, calculate EX2:

EX2 (02 × 1 12 × 6 22 × 15 32 × 20 42 × 15 52 × 6 62 × 1) / 32 240 / 32 7.5

Now, calculate VX:

VX 7.5 - 2.52 7.5 - 6.25 1.25

Conclusion

Using both theoretical formulas and step-by-step calculations, we have verified that the expectation (mean) and variance of the number of heads in 6 tosses of a fair coin are 2.5 and 1.25, respectively. This illustrates the utility and accuracy of the binomial distribution in modeling such scenarios.

Key Takeaways

The expectation (mean) of a binomial random variable is given by EX np. The variance of a binomial random variable is given by VX npq. These formulas can be verified through the listing of all possible outcomes and their corresponding frequencies.