Random Team Formation

Solution & Explanation

The problem of forming two equal teams randomly from six individuals can be seen as a probability problem involving the binomial distribution. Here's a step-by-step explanation of how we arrive at the expected number of rounds required to successfully form the teams:

Step 1: Understanding the Random Process

1. Random Decision: Each individual can choose to show either a face-up or face-down hand. This decision can be modeled as a Bernoulli trial with two possible outcomes: face-up or face-down, each with a probability of 0.5.

2. Objective: We need exactly three individuals to show a face-up hand and the remaining three to show a face-down hand. This ensures the formation of two equal teams.

Step 2: Binomial Distribution

Given that each person's decision is independent, the number of face-up hands, denoted by $F$, follows a binomial distribution:

• Parameters: $n = 6$ (number of trials), $p = 0.5$ (probability of success for each trial).

• Probability Mass Function: $P(F = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Calculate $P(F = 3)$: $P(F = 3) = \binom{6}{3} (0.5)^3 (0.5)^3 = 20 \times \left(\frac{1}{64}\right) = \frac{20}{64} = 0.3125$

Step 3: Geometric Distribution

The number of rounds $R$ needed to form the teams follows a geometric distribution because it represents the number of trials until the first success:

• Success Probability: $p = 0.3125$

• Expected Value: For a geometric distribution, the expected number of trials until the first success is given by: $E[R] = \frac{1}{p} = \frac{1}{0.3125} = 3.2$

Conclusion

The expected number of rounds required to form two equal teams randomly is 3.2. This means that, on average, it will take slightly more than three rounds for exactly three individuals to choose a face-up hand and the remaining three to choose a face-down hand, thereby successfully forming the teams.