What is the Probability Mass Function

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SUMMARY

The discussion focuses on calculating the probability that the first player wins a sudden-death chess match, where each game has a win probability of p for the first player and q for the second player. The Probability Mass Function (PMF) for the number of games played until the first player wins is derived as a geometric random variable, expressed as P(k) = (1-p)^(k-1) * p. Additionally, the discussion addresses the need to account for ties, represented by the probability of (1-p-q), and explores the implications of ties on the overall match duration and outcomes.

PREREQUISITES
  • Understanding of geometric random variables
  • Knowledge of probability theory, specifically PMFs
  • Familiarity with concepts of win/loss probabilities in games
  • Basic skills in solving equations involving probabilities
NEXT STEPS
  • Study the derivation of the geometric distribution in probability theory
  • Learn about the implications of ties in probability calculations
  • Explore advanced topics in stochastic processes related to game theory
  • Investigate the use of Markov chains in modeling game outcomes
USEFUL FOR

Mathematicians, statisticians, game theorists, and students studying probability and statistics, particularly those interested in modeling competitive scenarios and analyzing game outcomes.

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1. Homework Statement

Two world champion chess players play a sudden-death chess match where the first player to win a game wins the match. Each game is won by the first player with probability p and by the 2nd player with probability q and is a tie with probability (1-p-q).

a) what is the probability that the first player wins the match?
b) what is the Probability Mass Function, the mean, and the variance of the duration (number of games) of the match?


2. Homework Equations


3. The Attempt at a Solution

a) I think that this is a geometric random variable, seeing how many games (k) are needed for Wallace to win.
Px(k)= (1-p)^(k-1) p
yet I am not sure how to take into account that Wallace win FIRST, the games that come before him winning can only be ties.
does the equation of wallace NOT winning help at all?
= 1 - [ (1-q)^(k-1) q]
 
Physics news on Phys.org
- What is the chance that there is a tie?
- Then what is the chance that there are k - 1 ties in a row?
- What is the chance that there are k - 1 ties in a row and Wallace wins the kth game?
 

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