Calculating the Probability of Obtaining Three Heads and Two Tails When Tossing Five Fair Coins

Understanding probability is a fundamental concept in statistics, often applied in various fields, from finance to gambling. One classic example is the scenario of tossing fair coins. In this article, we will walk through a detailed explanation of how to determine the probability of obtaining three heads and two tails when tossing five fair coins. This process is a fantastic illustration of the binomial probability formula and its application.

Understanding the Problem

The problem at hand is: 'What is the probability of getting three heads and two tails when tossing five fair coins simultaneously?' This question can be tackled using the concept of combinations and the binomial probability formula. Let's break down the solution into manageable steps.

Step 1: Calculate the Total Number of Outcomes

When you toss a single fair coin, it can land on either heads (H) or tails (T). Therefore, for each coin toss, there are 2 possible results. Since we are tossing five coins, the total number of possible outcomes is:

[ 2^5 32 ]

This calculation reflects the fact that each coin toss is an independent event, and the total number of outcomes is the product of the number of outcomes for each coin.

Step 2: Calculate the Number of Favorable Outcomes

Next, we need to calculate the number of favorable outcomes where we get exactly three heads and two tails. This is where the binomial coefficient comes into play. The binomial coefficient, denoted as (binom{n}{k}), is used to calculate the number of ways to choose (k) successes out of (n) trials. In our case, (n 5) (total number of coin tosses), and (k 3) (number of heads).

[ binom{5}{3} frac{5!}{3!(5-3)!} frac{5!}{3!2!} frac{5 times 4}{2 times 1} 10 ]

Here, (5!) represents the factorial of 5, which is the product of all positive integers up to 5. Calculating the factorial of 5, (5 times 4 times 3 times 2 times 1 120), and similarly, (3! 6) and (2! 2).

Step 3: Calculate the Probability

Finally, to find the probability of getting exactly three heads and two tails, we divide the number of favorable outcomes by the total number of outcomes:

[ P(3 , heads , and , 2 , tails) frac{Number , of , favorable , outcomes}{Total , number , of , outcomes} frac{10}{32} frac{5}{16} ]

This step-by-step breakdown provides a clear understanding of how to apply the binomial probability formula to solve real-world problems. The probability of getting exactly three heads and two tails when tossing five fair coins is (frac{5}{16}).

Conclusion

The process of calculating the probability of obtaining a specific outcome using the binomial probability formula is a valuable skill. Understanding the step-by-step method detailed above can help in tackling similar problems in probability and statistics. Whether you are a student, a professional, or an enthusiast, mastering these concepts enhances your analytical and problem-solving skills.