Two Tails in Five Flips

Solution & Explanation

Understanding the Problem

The problem asks us to calculate the probability of getting exactly 2 tails when flipping a coin 5 times. This is a classic problem that can be solved using the concept of binomial distribution.

Key Concepts

1. Binomial Distribution: This is a probability distribution that summarizes the likelihood that a value will take one of two independent states and is defined by two parameters, n (number of trials) and p (probability of success on a single trial).

2. Combination: The number of ways to choose k successes in n trials is given by the combination formula $\binom{n}{k}$, which is calculated as: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

3. Probability of Success and Failure: In a fair coin toss, the probability of getting heads (success) or tails (failure) is $p = q = 0.5$.

Calculating the Probability

1. Total Possible Outcomes: When flipping a coin 5 times, each flip has 2 possible outcomes (heads or tails), so the total number of possible outcomes is: $2^5 = 32$

2. Using Binomial Distribution:

   • We want exactly 2 tails out of 5 flips, so:
     • $n = 5$ (number of trials)
     • $x = 2$ (number of tails)
     • $p = 0.5$ (probability of tails)
     • $q = 0.5$ (probability of heads)

3. Probability Formula: $P(X = 2) = \binom{5}{2} \times (0.5)^2 \times (0.5)^{5-2}$

   • Calculate $\binom{5}{2}$: $\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10$
   • Calculate $(0.5)^2 \times (0.5)^3$: $(0.5)^5 = 0.5^5 = \frac{1}{32}$

4. Final Probability: $P(X = 2) = 10 \times \frac{1}{32} = \frac{10}{32} = \frac{5}{16}$

Conclusion

The probability of getting exactly 2 tails in 5 coin flips is $\frac{5}{16}$ or approximately 31.25%. This solution uses the binomial distribution to calculate the probability, considering the combination of getting 2 tails in 5 trials and the probability of each outcome.