Calculating the Probability of Exactly 3 Heads in 10 Coin Tosses Using Binomial Distribution

The concept of probability plays a crucial role in a variety of fields, from sports and games to finance and data science. One classic example that helps in understanding probability is the tossing of a coin. In this article, we demonstrate the calculation of the probability of getting exactly 3 heads when a fair coin is tossed 10 times using the binomial distribution formula.

Understanding Binomial Distribution

When a fair coin is tossed, each toss has two possible outcomes: heads (H) or tails (T). The binomial distribution is a statistical concept used to determine the probability of k successes (in this case, getting heads) in n trials (coin tosses).

The Binomial Probability Formula

The binomial probability formula is given by:

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

Where:

P(X k) is the probability of getting exactly k successes (heads) in n trials (coin tosses) n is the number of trials (10 coin tosses) k is the number of successful outcomes (3 heads) nCk is the binomial coefficient, calculated as n! / (k!(n - k)!) p is the probability of success (getting heads) in a single trial, which is 0.5 for a fair coin 1 - p is the probability of failure (getting tails) in a single trial, also 0.5 for a fair coin

Calculating the Probability

Let's calculate the probability of getting exactly 3 heads in 10 coin tosses using the binomial probability formula:

P(X 3) 10C3 (0.5)^3 (0.5)^{10 - 3}

In simpler terms, we can break this down as follows:

First, calculate the binomial coefficient, 10C3, which is the number of ways to choose 3 items from 10: 10C3 10! / (3! (10 - 3)!) 120 Next, calculate the probability of getting 3 heads and 7 tails: P(X 3) 120 cdot (0.5)^3 cdot (0.5)^7 Finally, compute the result: P(X 3) 120 cdot 0.125 cdot 0.0078125 0.1171875

The probability of getting exactly 3 heads in 10 coin tosses is therefore 0.1171875, or approximately 11.72%.

Binomial Probability in Context

Without using the formula, if we consider a simpler approach:

When a coin is tossed 10 times, there are 2^{10} 1024 possible outcomes. The number of ways to get exactly 3 heads out of 10 is given by the combination formula 10C3, which equals 120. Thus, the probability of getting exactly 3 heads is:

P(X 3) 120 / 1024 0.11719

Conclusion

Understanding and applying the binomial probability formula is essential in numerous scenarios, from everyday coin tosses to complex statistical analyses. This method not only helps in calculating the probabilities but also provides a deeper understanding of probability theory.