Coin Toss : stats percentage probability

Question

In the group game “Odd-One Out”: All players take out a fair coin, and everyone flips at the same time. If one player (the “odd-one”) receives heads and everyone else receives tails (or vice-versa), the game ends. Otherwise the game goes on for more rounds, until an “odd-one” is finally found. In a group of 12 players, what is the percentage probability that the game will:

A) End in the 1st round?

B) Not end in the first 2 rounds?

C) End in the 3rd round?

Answer 1

For 12 simultaneous coin flips by 12 people, the number of possible outcomes (size of the sample space) is 2¹² = 4096. Twelve (12) of these outcomes have 1 head and 11 tails, since each of the 12 people could be the ‘odd one out’. Likewise, there are 12 outcomes with 1 tail and 11 heads. So the total number of possible ‘successes’ is 24.

A) For the game to end in a single round, the probability is that of a single success in one trial:

24/4096 = 0.0059 = 0.59%.

B) For the game to not end after two rounds, we must have two non-successes in a row. Each non-success has a probability of 1 – 24/4096 . Two in a row (we assume all coin tosses are independent) have a probability of (1 – 24/4096)² = 0.9883 = 98.83%.

C) For the game to end after exactly three rounds, we must have two non-successes, followed by one success. The probability of this is the product of (A) and (B), or (1 – 24/4096)² · 24/4096 = 0.0058 = 0.58%.