The Various Outcomes of Flipping a Coin 10 Times to Get Exactly Two Heads

Understanding the binomial distribution is crucial when dealing with binary outcomes, such as the various ways a coin can land when flipped multiple times. This article will explore the number of possible outcomes for flipping a coin 10 times and landing exactly 2 heads, diving into the mathematical concepts, calculations, and the significance of the binomial distribution in this context.

Introduction to Coin Flipping and Binary Outcomes

Coin flipping is a classic example of a binary outcome. Each flip can result in either heads (H) or tails (T). Hence, for 10 flips, the total number of possible outcomes is 210, which equals 1024. Each sequence of outcomes from HHHHHHHHHH to TTTTTTTTTT represents a unique combination of heads and tails.

Calculating the Number of Outcomes with Exactly Two Heads

Given 10 coin flips, the problem is to calculate the number of possible outcomes that contain exactly two heads. Mathematically, this can be represented using the binomial coefficient 10 choose 2, which is calculated as:

10! / (2!8!) 45

This formula, nCk??n! / (k!(n-k)!), is used in combinatorial mathematics to find the number of ways to choose k successes out of n trials.

Further Exploration of Coin Flipping Outcomes

Let's delve deeper into the distribution of outcomes for different numbers of heads in 10 coin flips:

0 heads (or 10 tails): 1 outcome 1 head (or 9 tails): 10 outcomes 2 heads (or 8 tails): 45 outcomes 3 heads (or 7 tails): 120 outcomes 4 heads (or 6 tails): 210 outcomes 5 heads (or 5 tails): 252 outcomes 6 heads (or 4 tails): 210 outcomes 7 heads (or 3 tails): 120 outcomes 8 heads (or 2 tails): 45 outcomes 9 heads (or 1 tail): 10 outcomes 10 heads (or 0 tails): 1 outcome

The distribution is symmetrical, with the highest probability at 5 heads (5 tails).

Binomial Distribution and Probability

The binomial distribution is a probability distribution that models the number of successes in a fixed number of independent Bernoulli trials with the same probability of success. For a fair coin, the probability of heads on any single flip is 0.5. Using the binomial probability mass function (PMF):

Pr(X k) binom{n}{k} p^k (1-p)^{n-k}

Substituting n10, k2, and p0.5 into the formula, we get:

binom{10}{2} (0.5)^2 (1-0.5)^{10-2} frac{10!}{2!8!} * (0.5)^2 * 0.5^8 45 * 0.25 * 0.00390625 0.0439453125

This probability is approximately 4.39%, which is the chance of getting exactly 2 heads in 10 coin flips.

Conclusion

In summary, the binomial distribution is a powerful tool for understanding and calculating the probability of binary outcomes, such as the number of heads in multiple coin flips. The calculations and analysis demonstrate the symmetrical nature of the outcomes and the underlying probabilities involved. Understanding these concepts is essential in fields like statistics, probability theory, and even game theory.