Obtaining Three Heads

Solution & Explanation

To solve the problem of finding the probability of obtaining exactly three heads when flipping a coin four times, we can use the principles of probability and combinatorics. Let's break down the solution step-by-step:

Step 1: Understanding the Total Number of Outcomes

• Coin Flip Outcomes: Each flip of a fair coin has two possible outcomes: heads (H) or tails (T).

• Total Flips: Since we are flipping the coin four times, the total number of possible outcomes is calculated as:

  $2^4 = 16$

Step 2: Identifying Favorable Outcomes

• Desired Outcome: We want exactly three heads in four flips.
• Combinations for Three Heads: To achieve exactly three heads, we can have one tail in any of the four positions. This means we need to count the number of ways to arrange three heads and one tail in four positions.
• Possible Arrangements: The favorable outcomes are:
  • HHHT
  • HHTH
  • HTHH
  • THHH
• Number of Favorable Outcomes: There are 4 ways to arrange three heads and one tail.

Step 3: Calculating the Probability

• Probability Formula: The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.

  $P(3 \text{ Heads}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}$

• Calculation:

  $P(3 \text{ Heads}) = \frac{4}{16} = \frac{1}{4} = 0.25$

Step 4: Verification Using Binomial Distribution

• Binomial Distribution: The problem can also be viewed as a binomial distribution where:

  • Number of trials $n = 4$
  • Probability of success (getting a head) $p = 0.5$
  • Number of successes $k = 3$

• Binomial Probability Formula:

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

• Applying Values:

  $P(X = 3) = \binom{4}{3} \cdot (0.5)^3 \cdot (0.5)^{4-3}$

  $P(X = 3) = 4 \cdot \left(\frac{1}{2}\right)^4 = 4 \cdot \frac{1}{16} = \frac{1}{4} = 0.25$

Conclusion

Both the combinatorial approach and the binomial distribution method confirm that the probability of getting exactly three heads in four coin flips is 0.25 or 25%. This demonstrates the consistency between basic probability principles and statistical distributions.