In the popular game Plinko on The Price Is Right,...

In the popular game Plinko on The Price Is Right, contestants drop a circular disk (a "chip") down a pegged board; the chip bounces down the board and lands in a slot corresponding to one of five dollar mounts. The random variable $X=$ winnings from one chip dropped from the middle slot has roughly the following distribution. $$\begin{array}{c|ccccc}{x} & {\$ 0} & {\$ 100} & {\$ 500} & {\$ 1000} & {\$ 10,000} \\ \hline p(x) & {.39} & {.03} & {.11} & {.24} & {.23}\end{array}$$ (a) Graph the probability mass function of $X .$ (b) What is the probability a contestant makes money on a chip? (c) What is the probability a contestant makes at least $\$ 1000$ on a chip? (d) Determine the expected winnings. Interpret this number. (e) Determine the corresponding standard deviation.

In the popular game Plinko on The Price Is Right, contestants drop a circular disk (a "chip") down a pegged board; the chip bounces down the board and lands in a slot corresponding to one of five dollar mounts. The random variable $X=$ winnings from one chip dropped from the middle slot has roughly the following distribution.
$$\begin{array}{c|ccccc}{x} & {\$ 0} & {\$ 100} & {\$ 500} & {\$ 1000} & {\$ 10,000} \\ \hline p(x) & {.39} & {.03} & {.11} & {.24} & {.23}\end{array}$$
(a) Graph the probability mass function of $X .$
(b) What is the probability a contestant makes money on a chip?
(c) What is the probability a contestant makes at least $\$ 1000$ on a chip?
(d) Determine the expected winnings. Interpret this number.
(e) Determine the corresponding standard deviation.

Key Concepts

Expected Value (Mean) of a Discrete Random Variable

The expected value is a measure of the central tendency of a probability distribution. It is calculated as the sum of the products of each outcome value and its corresponding probability. The expected value provides a long-run average value of the random variable over many repetitions of the experiment.

Calculating Event Probabilities Using a PMF

Determining the probability of a specific event when given a PMF often involves summing the probabilities of all outcomes that satisfy the event's conditions. This method is especially useful when the event is defined in terms of a range or a condition on the outcomes of the discrete random variable.

Variance and Standard Deviation

Variance measures the spread of the random variable’s outcomes around the expected value, quantifying the degree of variability or dispersion in the distribution. The standard deviation, which is the square root of the variance, provides a scale for this dispersion in the same units as the random variable, offering a more intuitive measure of variability.

Probability Mass Function (PMF)

A probability mass function gives the probabilities of occurrence of all possible discrete outcomes for a random variable. It is a fundamental tool in probability theory that assigns a probability, which is a number between 0 and 1, to each value in the variable’s sample space, ensuring that the total probability sums to 1.

Graphical Representation of a PMF

Graphing a PMF involves plotting the discrete outcomes on the horizontal axis and their corresponding probabilities on the vertical axis. This visual representation helps to clearly illustrate the distribution of probability across the outcomes and makes it easier to understand the behavior of the random variable.

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