Probability of Obtaining at Most 2 Heads in 20 Coin Flips

In this article, we will explore the probability of obtaining at most 2 heads when flipping a fair coin 20 times. We will use the binomial distribution to calculate this probability, providing a comprehensive understanding of the underlying mathematical principles.

Understanding the Binomial Distribution

To begin with, let's understand the binomial distribution. This probability distribution describes the number of successes in a sequence of n independent Bernoulli trials, where each trial has two possible outcomes: success (heads) or failure (tails). In our example, flipping a fair coin can be modeled as a Bernoulli trial with a success probability of 0.5 for heads.

Calculating the Probability

We need to find the probability of obtaining at most 2 heads in 20 coin flips. This can be expressed as . We can break this down into the probability of obtaining exactly 0, 1, or 2 heads.

Step 1: Calculate Each Probability

The probability of obtaining exactly k heads in n flips is given by the binomial probability formula:

For our scenario, where n 20 and p 0.5, we calculate:

For k 0:

The probability of getting 0 heads is

For k 1:

The probability of getting 1 head is

For k 2:

The probability of getting 2 heads is

Step 2: Sum the Probabilities

Now, we sum the probabilities to find :

Step 3: Calculate the Final Probability

The final probability of getting at most 2 heads when flipping a coin 20 times is approximately:

This is approximately 0.0201, or 2.01%.

Conclusion

The probability of obtaining at most 2 heads in 20 coin flips is relatively low, at approximately 0.0201. This demonstration helps us understand how the binomial distribution applies to real-world scenarios and provides valuable insights into probability calculations.