Question

In a holiday gathering, two families: family 1 and family 2 play a game repeatedly and independently until one of them win. A given game ends in a tie with probability of 0.2 . The probability that family 1 wins an individual game is 0.35 , while probability that family 2 wins an individual game is 0.45 . What is the probability that family 1 is eventual winner.

Answer 1

To find the probability that family 1 is the eventual winner, we can use the concept of a geometric distribution. The probability that family 1 is the eventual winner is approximately 0.486.

Step-by-step explanation:

To find the probability that family 1 is the eventual winner, we can use the concept of a geometric distribution. Let X be the number of games played until family 1 wins. Since the game ends in a tie with a probability of 0.2, the probability of family 1 winning an individual game is 0.35, and the probability of family 2 winning an individual game is 0.45, the probability of family 1 winning is:

P(X > k) = (0.35 * 0.45)^k * 0.35

Summing up this infinite geometric series:

P(X > 0) = 1 + (0.35 * 0.45) + (0.35 * 0.45)^2 + (0.35 * 0.45)^3 + ...

The formula for the sum of an infinite geometric series is:

S = a / (1 - r), where a is the first term and r is the common ratio.

Substituting the values, the probability that family 1 is the eventual winner is approximately 0.486.