Calculating Probability Exactly 5 Heads In 10 Coin Flips

In probability theory, the binomial distribution is a fundamental concept that describes the probability of obtaining a certain number of successes in a sequence of independent experiments, each with the same probability of success. This distribution is widely used in various fields, including statistics, finance, and engineering, to model and analyze phenomena involving discrete outcomes.

The binomial distribution is characterized by two parameters: n, the number of trials or experiments, and p, the probability of success in a single trial. The probability of obtaining exactly k successes in n trials is given by the following formula:

P(X = k) = \binom{n}{k} * p^k * (1 - p)^(n - k)

where:

• P(X = k) is the probability of getting exactly k successes
• $\binom{n}{k}$ is the binomial coefficient, which represents the number of ways to choose k successes from n trials. It is calculated as $\frac{n!}{k!(n-k)!}$, where ! denotes the factorial function.
• p is the probability of success on a single trial
• (1 - p) is the probability of failure on a single trial
• n is the number of trials
• k is the number of successes

To fully grasp the essence of the binomial probability formula, let's delve into its components and their significance. First, the binomial coefficient, denoted as $\binom{n}{k}$, plays a crucial role in determining the number of ways to achieve exactly k successes out of n trials. This coefficient accounts for all possible combinations of successes and failures, ensuring that we consider every distinct scenario. For instance, if we're flipping a coin 10 times and want to find the probability of getting exactly 5 heads, the binomial coefficient will tell us how many different ways we can arrange those 5 heads within the 10 flips.

Next, we have p, which represents the probability of success on a single trial. This probability is a key factor in determining the overall likelihood of obtaining a specific number of successes. In our coin-flipping example, p would be the probability of getting a head on a single flip, which is typically 0.5 for a fair coin. The term p raised to the power of k, denoted as p^k, calculates the probability of getting k successes in a row.

Similarly, (1 - p) represents the probability of failure on a single trial. This is simply the complement of the probability of success. In the coin-flipping scenario, (1 - p) would be the probability of getting a tail on a single flip, which is also 0.5 for a fair coin. The term (1 - p) raised to the power of (n - k), denoted as (1 - p)^(n - k), calculates the probability of getting (n - k) failures in a row.

Finally, by multiplying the binomial coefficient, the probability of k successes (p^k), and the probability of (n - k) failures ((1 - p)^(n - k)), we arrive at the binomial probability of getting exactly k successes in n trials. This formula provides a powerful tool for analyzing a wide range of scenarios where we need to calculate the likelihood of specific outcomes in a series of independent events.

The Problem: Probability of Exactly 5 Heads in 10 Coin Flips

Now, let's apply this formula to solve the specific problem at hand: What is the probability of getting exactly 5 heads in 10 coin flips?

In this case, we have:

• n = 10 (number of coin flips)
• k = 5 (number of heads we want to get)
• p = 0.5 (probability of getting a head on a single flip, assuming a fair coin)

Plugging these values into the formula, we get:

P(X = 5) = \binom{10}{5} * (0.5)^5 * (1 - 0.5)^(10 - 5)

Let's break down the calculation step by step:

1. Calculate the binomial coefficient:

   $\binom{10}{5} = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!} = \frac{10 * 9 * 8 * 7 * 6}{5 * 4 * 3 * 2 * 1} = 252$

   The binomial coefficient $\binom{10}{5}$ tells us that there are 252 different ways to arrange 5 heads within 10 coin flips. This accounts for all the possible sequences of heads and tails that result in exactly 5 heads.

2. Calculate the probability of 5 heads:

   (0. 5)^5 = 0.03125

   The probability of getting 5 heads in a row is 0.03125. This is a relatively small probability, as it represents just one specific sequence of 5 heads and 5 tails.

3. Calculate the probability of 5 tails:

   (1 - 0. 5)^(10 - 5) = (0.5)^5 = 0.03125

   Similarly, the probability of getting 5 tails in a row is also 0.03125. This reflects the symmetry of the situation, as getting 5 tails is just as likely as getting 5 heads.

4. Multiply all the terms together:

   P(X = 5) = 252 * 0.03125 * 0.03125 = 0.24609375

   By multiplying the binomial coefficient, the probability of 5 heads, and the probability of 5 tails, we arrive at the final probability of getting exactly 5 heads in 10 coin flips, which is approximately 0.246.

Therefore, the probability of getting exactly 5 heads in 10 coin flips is approximately 0.246, or about 24.6%. Now, let's express this probability as a fraction to match the answer choices provided.

To convert 0.24609375 to a fraction, we can write it as 24609375 / 100000000. We can simplify this fraction by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it.

In this case, the GCD of 24609375 and 100000000 is 390625. Dividing both the numerator and denominator by 390625, we get:

24609375 / 390625 = 63

100000000 / 390625 = 256

So, the simplified fraction is 63 / 256.

The Answer

Therefore, the probability of getting exactly 5 heads in 10 coin flips is 63/256, which corresponds to option B.

Conclusion

In conclusion, the binomial probability formula provides a powerful tool for calculating the likelihood of obtaining a specific number of successes in a sequence of independent trials. By understanding the components of the formula and applying it systematically, we can solve a wide range of probability problems, including the classic example of coin flips. In this case, we determined that the probability of getting exactly 5 heads in 10 coin flips is 63/256, demonstrating the practical application of the binomial distribution in analyzing real-world scenarios.