Three Tails

Solution & Explanation

To solve the problem of finding the probability that exactly three coins show tails given that at least two of them land on tails, we need to analyze the possible outcomes of tossing four coins.

Step-by-Step Breakdown:

1. Total Outcomes:

   • When tossing four coins, each coin has two possible outcomes: heads (H) or tails (T).
   • Hence, the total number of possible outcomes is $2^4 = 16$.

2. Outcomes with at Least Two Tails:

   • We are given that at least two coins land on tails. Therefore, we must exclude outcomes where there are fewer than two tails:
     • Outcomes with 0 tails: $\{HHHH\}$
     • Outcomes with 1 tail: $\{HHHT, HHTH, HTHH, THHH\}$
   • Excluding these, we have: $16 - (1 + 4) = 11$ outcomes where there are at least two tails.

3. Outcomes with Exactly Three Tails:

   • We need to find the number of outcomes where exactly three coins show tails.
   • Possible outcomes with exactly three tails are:
     • $\{TTTH, TTHT, THTT, HTTT\}$
   • There are 4 such outcomes.

4. Calculating the Probability:

   • The probability that exactly three coins show tails given that at least two coins show tails is calculated as:

     $P(\text{3 tails | at least 2 tails}) = \frac{\text{Number of favorable outcomes (exactly 3 tails)}}{\text{Total outcomes with at least 2 tails}}$

     $P(\text{3 tails | at least 2 tails}) = \frac{4}{11}$

5. Conclusion:

   • The probability that exactly three coins show tails given that at least two of them land on tails is $\frac{4}{11}$.

This solution demonstrates the application of basic probability principles, involving counting outcomes and conditional probability. It highlights the importance of understanding the sample space and using logical reasoning to filter outcomes based on given conditions.