In a game of skill, a player has probability 1/3,5/12,1/4 of
scoring 0,1,2 respectively and the game ends when he scores a zero.
Assuming independence of trials, find the probability of the total
score being 'n' when the game ends?
To find the probability of the total score being 'n' when the game ends, we can consider the different ways in which this can happen.
Since the game ends when the player scores a zero, we need to consider all the possible sequences of scores that can lead to a total score of 'n'. Let's break down the problem using conditional probability:
1. If the total score is 'n', the last score must be zero. The probability of scoring a zero is 1/3.
2. Since the game ends when a zero is scored, the previous scores can be any combination of the remaining scores 1 and 2. We need to calculate the probability of each combination leading to the remaining score of 'n-0' (since we already have a zero). Let's denote this probability as P(n-0).
3. The probability of scoring a 1 is 5/12, and the probability of scoring a 2 is 1/4.
4. For each previous score, we can calculate the remaining score by recursively using step 2 and step 3 until we reach a total score of zero.
Now, let's use these steps to calculate the probability of the total score being 'n':
P(n) = P(n-0) * (1/3)
To calculate P(n-0), we need to consider the two possible cases:
a) If the previous score was 1:
P(n-0) = P(n-1) * (5/12)
b) If the previous score was 2:
P(n-0) = P(n-2) * (1/4)
Using these recursive steps, we can calculate the probability P(n) for any given value of 'n'.