What is the Probability Mass Function

[janela]

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]

 

[CompuChip]

- 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?

 

[tacman]

The Probability Mass Function (PMF) is a function that gives the probability of a discrete random variable taking on a specific value. In this case, the random variable is the number of games played in the chess match and the values it can take on are the different numbers of games (1, 2, 3, etc.). The PMF for this situation would be:

P(X = k) = (1-p-q)^(k-1) * p, for k = 1, 2, 3, ...

This means that the probability of the first player winning the match in exactly k games is equal to the probability of (k-1) ties followed by a win by the first player.

a) The probability that the first player wins the match can be calculated by adding up the probabilities of winning in 1 game, 2 games, 3 games, etc. In other words, it is the sum of the PMF over all possible values of k. This can be written as:

P(first player wins match) = ∑ [(1-p-q)^(k-1) * p], for k = 1, 2, 3, ...

b) The mean of the duration of the match is given by the expected value of the random variable X, which can be calculated using the following formula:

E(X) = ∑ k * P(X = k), for k = 1, 2, 3, ...

The variance of the duration of the match is given by:

Var(X) = ∑ (k - E(X))^2 * P(X = k), for k = 1, 2, 3, ...

In this case, the mean and variance can be calculated using the PMF from part a.

It is important to note that the PMF, mean, and variance are all specific to this particular situation and may be different for other situations with different probabilities for winning and tying.