In a chess tournament, n games are being played,...

In a chess tournament, $n$ games are being played, independently. Each game ends in a win for one player with probability $0.4$ and ends in a draw (tie) with probability $0.6$. Find the PMFs of the number of games ending in a draw, and of the number of players whose games end in draws.

Key Concepts

Indicator Random Variables

Indicator random variables take the value 1 if a certain event occurs and 0 otherwise. They simplify the analysis of counting problems by converting events into binary outcomes. Summing indicator variables over multiple independent trials gives a pathway to apply the binomial model, particularly useful in transforming counts from one domain (games) to another (players).

Transformation of Random Variables

Often, the quantity of interest is not directly given by the basic outcomes but as a function of those outcomes. In these cases, techniques for transforming random variables are used to derive the distribution of the resulting variable (for example, translating the count of games with draws into the count of players who participate in those draws).

Bernoulli Trials

A Bernoulli trial is an experiment with exactly two possible outcomes, often labeled as success (with a certain probability) and failure (with the complementary probability). When the experiment (or game, in abstract terms) is repeated independently, the series of Bernoulli trials forms the basis for the binomial distribution.

Binomial Distribution

This is a discrete probability distribution describing the number of successes in a fixed number of independent trials, where each trial has two outcomes (commonly called success and failure) with constant probability. It is used when modeling the total count of a particular outcome (such as a draw) from several independent events.

Probability Mass Function (PMF)

The PMF of a discrete random variable specifies the probability that the variable is exactly equal to a given value. In problems involving counts, such as the number of games ending in a draw, the PMF provides a complete description of the distribution by outlining the probabilities of all possible outcomes.

Transcript

00:01 This scenario described for this exercise is somebody playing a video game in which they confront opponents and have an 80 % probability of defeating each one.

00:10 They keep playing until they are defeated by their opponent.

00:15 And so the probability of success is always 80%, or the probability of defeating the opponent is always 80 % and it's independent of previous encounters.

00:23 So basically what's being described are brunei trials.

00:28 And so the game goes on until the player is defeated.

00:32 So let's call the probability p of success 0 .2.

00:39 So that is success is the player being defeated by their opponent.

00:46 And let's describe the random variable x as the number of opponents required to play against before being defeated.

00:59 So basically the number of trials until a success.

01:10 And so x is a geometric random variable.

01:13 And so for part a, we are asked, what is the probability mass function of the number of opponents contested in a game? so that is simply the probability mass function for a geometric random variable.

02:13 Now, for part b, we are asked for the probability that the player defeats at least two opponents in the game, which means that it takes at least three trials to face defeat.

02:25 That is, we need at least three trials for the success.

02:29 So we're looking for the probability that x is at least 3, which is the 1 minus the probability that x is less than equal to 2.

02:51 So for the probability of x equals 1, that is simply p.

03:00 And the probability that x equals 2 is 1 minus 0 .2 times 0 .2.

03:17 So this is 1 minus 0 .36, which comes out to 0 .64.

03:28 Now for part c, we are asked for the expected number of components contested in the game.

03:44 So this is the expected number of trials until the success.

03:55 This is given by 1 over p.

04:07 So the expected number of opponents faced before losing is 5.

04:19 For part d, we are asked for the probability that the player contests four or more components in a game...