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 need to consider the different ways in which the player can achieve a total score of 'n'.
Let's break it down step by step:
1. Consider the scenario where the total score is 'n' after just one round of the game. In this case, the player must have scored 'n' points and the game must have ended. Since the game ends when the player scores a zero, the probability of the total score being 'n' after one round is equal to the probability of the player scoring 'n' points in one round.
2. Now, consider the scenario where the total score is 'n' after two rounds of the game. In this case, the player could have scored 'n' points in the first round and then zero points in the second round, or the player could have scored 'n-1' points in the first round and '1' point in the second round, and so on. We need to consider all such possible combinations and calculate their probabilities.
3. Similarly, repeat the above process for three rounds, four rounds, and so on until we reach a point where the total score is less than 'n'.
4. Finally, we sum up all the probabilities obtained in step 1, 2, 3 for each respective scenario to get the total probability of the total score being 'n' when the game ends.
Writing a mathematical expression for the total probability of the total score being 'n' might be challenging due to the various combinations to consider in steps 2 and 3. It would be easier to solve this problem numerically using a computer program or a spreadsheet.
Would you like me to guide you on how to solve this problem numerically using a spreadsheet or a computer program?