What does $X$ represent in the three-dice experiment?

$X$ is the number of the three dice that show the single number chosen before the roll. Its possible values are $0,1,2,$ and $3$.

There are three independent trials, each die either matches the chosen number or does not, and each die has the same match probability $1/6$. Thus $X\sim B(3,1/6)$.

What is the difference between probability and relative frequency?

A theoretical probability is calculated from the model. A relative frequency is the observed count of an outcome divided by the total number of trials.

What is the theoretical mean number of matches?

$\mu_X=np=3(1/6)=0.5$. It describes the long-run average number of matches per game, not the outcome guaranteed in a single game.

How is expected profit determined?

First use the payout schedule to find profit $g(x)$ for each number of matches. Then calculate $\mu_Y=\sum_{x=0}^{3}g(x)P(X=x)$.

ProbabilityLesson 2 of 2 in the course Discrete Random Variables

Binomial Match Counts and Empirical Results for Three Dice

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

Save course

Lesson progress

0 / 5

Your progress is kept in this browser only.Sign up to save it

Summary

In this lesson, you will model a game in which a player chooses a number and rolls three regular six-sided dice. You will compare the theoretical probability model with results collected through repeated trials and learn how the number of matches relates to profit.

Choose one number, such as 4, before rolling the three dice. Let $X$ be the number of dice that show the chosen number. Since there are three dice, $X$ can take the values $0, 1, 2,$ or $3$ .

All flashcards

What does $X$ represent in the three-dice experiment?: $X$ is the number of the three dice that show the single number chosen before the roll. Its possible values are $0,1,2,$ and $3$ .
Why is $X$ binomial?: There are three independent trials, each die either matches the chosen number or does not, and each die has the same match probability $1/6$ . Thus $X\sim B(3,1/6)$ .
What is the difference between probability and relative frequency?: A theoretical probability is calculated from the model. A relative frequency is the observed count of an outcome divided by the total number of trials.
What is the theoretical mean number of matches?: $\mu_X=np=3(1/6)=0.5$ . It describes the long-run average number of matches per game, not the outcome guaranteed in a single game.
How is expected profit determined?: First use the payout schedule to find profit $g(x)$ for each number of matches. Then calculate $\mu_Y=\sum_{x=0}^{3}g(x)P(X=x)$ .

Quiz questions

What binomial model describes the number of matches when three fair dice are rolled for one chosen number?Answer: $X\sim B(3,1/6)$
What is the probability of getting exactly two matches?Answer: $15/216\approx0.0694$
How is the relative frequency of exactly one match calculated from observed trials?Answer: Divide the number of trials with exactly one match by the total number of trials.
What does the theoretical mean $\mu_X=0.5$ describe?Answer: The long-run average number of matches per game under the model.
Why can a class relative frequency differ from the theoretical probability?Answer: Observed results vary from sample to sample, particularly when the number of trials is limited.

Based on Introductory Statistics 2e by Barbara Illowsky, Susan Dean (OpenStax), licensed CC BY 4.0. Adapted into a summary, flashcards and quizzes.

All Probability lessons

ProbabilityMultiplication and Addition Rules for Two Events5 flashcards · 5 quiz questions · 4 gamesPlay free →ProbabilityGeometric Distributions: Counting Conventions and Waiting-Time Formulas5 flashcards · 5 quiz questions · 4 gamesPlay free →ProbabilityProbability of Two Draws: Replacement, Conditional Events, and Frequency5 flashcards · 5 quiz questions · 4 gamesPlay free →

Summary

All flashcards

Quiz questions

More in Probability