Two Heads and One Tail

Solution & Explanation

When flipping a fair coin three times, there are a total of $2^3 = 8$ possible outcomes. These outcomes are:

• HHH
• HHT
• HTH
• HTT
• THH
• THT
• TTH
• TTT

Given the problem, we need to determine the probability of obtaining exactly two heads and one tail, regardless of the order. The combinations that satisfy this condition are:

• HHT
• HTH
• THH

Thus, there are 3 favorable outcomes.

Probability Calculation

The probability of a specific outcome when flipping a fair coin is $\frac{1}{2}$ for heads and $\frac{1}{2}$ for tails. To find the probability of getting exactly two heads and one tail, we use the binomial probability formula:

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

Where:

• $n = 3$ (number of trials)
• $k = 2$ (number of successes, i.e., heads)
• $p = 0.5$ (probability of getting heads on each flip)
• $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.

Step-by-Step Calculation:

1. Compute the binomial coefficient $\binom{3}{2}$: $\binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \cdot 2 \cdot 1}{2 \cdot 1 \cdot 1} = 3$

2. Compute the probability using the binomial formula: $P(X = 2) = \binom{3}{2} \cdot (0.5)^2 \cdot (0.5)^{3-2}$ $P(X = 2) = 3 \cdot (0.5)^2 \cdot (0.5)^1 = 3 \cdot 0.25 \cdot 0.5 = 3 \cdot 0.125 = 0.375$

Conclusion:

The probability of obtaining exactly two heads and one tail when flipping a coin three times is 0.375 or 37.5%.

This solution aligns with the principles of binomial distribution, where each coin flip is an independent trial with two possible outcomes, and we are interested in the number of successes (heads) in these trials.