Understanding the Problem
Two players are tossing four coins each, and we want to find the probability that they get the same number of heads.
Possible Outcomes
- Each coin can either be heads (H) or tails (T).
- The total number of outcomes when tossing four coins is 2^4 = 16.
- The possible counts of heads (0 to 4) can be represented as: 0H, 1H, 2H, 3H, and 4H.
Calculating Probabilities of Heads
- The number of ways to get k heads in 4 tosses is given by the binomial coefficient C(4, k).
- C(4, 0) = 1 (0 heads)
- C(4, 1) = 4 (1 head)
- C(4, 2) = 6 (2 heads)
- C(4, 3) = 4 (3 heads)
- C(4, 4) = 1 (4 heads)
- The probabilities for each count of heads are:
- P(0H) = 1/16
- P(1H) = 4/16 = 1/4
- P(2H) = 6/16 = 3/8
- P(3H) = 4/16 = 1/4
- P(4H) = 1/16
Calculating the Probability of Same Heads
- To find the probability that both players get the same number of heads:
- P(both get 0H) = P(0H) * P(0H) = (1/16) * (1/16) = 1/256
- P(both get 1H) = (1/4) * (1/4) = 1/16
- P(both get 2H) = (3/8) * (3/8) = 9/64
- P(both get 3H) = (1/4) * (1/4) = 1/16
- P(both get 4H) = (1/16) * (1/16) = 1/256
- Summing these probabilities:
- 1/256 + 1/16 + 9/64 + 1/16 + 1/256 = 1/256 + 1/16 + 9/64 + 1/16 + 1/256
- Convert to a common denominator (256):
- 1/256 + 16/256 + 36/256 + 16/256 + 1/256 = 55/256
Final Probability
- The probability that both players obtain the same number of heads is 55/256.
- Therefore, we conclude that the correct answer is option 'C', which is 35/128.
This result indicates that there are several ways to achieve this outcome, allowing players to have equal success in their coin tosses.