Ending the Coin Tossing Game in One Round: Probabilities and Scenarios

Suppose each of three persons tosses a coin. If the outcome of one of the tosses differs from the other outcomes, then the game ends. If not, then the persons start over and re-toss their coins. Assuming fair coins, what is the probability that the game will end with the first round of tosses? If all three coins are biased and have probability 1/4 of landing heads, what is the probability that the game will end at the first round? 

In this game, three persons each toss a fair coin. The game ends when one of the persons gets a different outcome than the others, that is, when two of the persons get heads and one gets tails, or vice versa. If the outcomes are the same, the persons start over and re-toss their coins. The question is, what is the probability that the game will end with the first round of tosses?

To solve this problem, we can use the concept of conditional probability. The probability that the game will end with the first round of tosses is the sum of the probabilities of each possible combination of outcomes that results in the game ending.

The probability of getting two heads and one tail, for example, is (1/2) * (1/2) * (1/2) = 1/8. Similarly, the probability of getting two tails and one head is also 1/8. The total probability of the game ending with the first round of tosses is therefore (1/8) + (1/8) = 1/4.

Now, if all three coins are biased and have a probability of 1/4 of landing heads, the probability of the game ending in the first round is (3/4) * (3/4) * (3/4) = 27/64.

So, if the coins are fair, the probability of the game ending in the first round is 1/4. If the coins are biased, the probability of the game ending in the first round is 27/64.

To summarize,

In this game, three persons each toss a fair coin and the game ends when one of the persons gets a different outcome than the others.

To solve this problem, we can use the concept of conditional probability and sum the probabilities of each possible combination of outcomes that results in the game ending.

The probability of the game ending in the first round if the coins are fair is 1/4.

If the coins are biased and have a probability of 1/4 of landing heads, the probability of the game ending in the first round is 27/64.