/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 162 The Big Six Wheel (or Wheel of F... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 162

The Big Six Wheel (or Wheel of Fortune) is a casino and carnival game that is well known for being a big money maker for the casinos. The wheel has 54 equally likely slots (outcomes) on it. The slot that pays the largest amount of money is called the joker. If a player bets on the joker, the probability of winning is \(1 / 54\). The outcome of any given play of this game (a spin of the wheel) is independent of the outcomes of previous plays. a. Find the probability that a player who always bets on joker wins for the first time on the 15 th play of the game. b. Find the probability that it takes a player who always bets on joker more than 70 plays to win for the first time.

Short Answer

Expert verified
a. The probability that a player who always bets on joker wins for the first time on the 15 th play of the game is given by substituting appropriate values in the formula, which is roughly 0.0305. b. The probability that it takes a player who always bets on joker more than 70 plays to win for the first time is calculated by subtracting the sum of probabilities from 1 until the 69th trial from 1, which is roughly 0.2331.

Step by step solution

01

Understand the formula

The probability mass function of the geometric distribution is expressed as: \[ P(X = k) = p*q^(k-1) \], where p is the probability of success, q is the probability of failure, and k is the count of trials. The probability of failure q is 1-p.
02

Solve for Part a

We apply the formula to our case: The probability of success p is \(1 / 54\), the number of plays until a win occurs k is 15. We get: \[ P(X = 15) = (1/54)*(53/54)^(15-1) \]
03

Solve for Part b

We now want to find the cumulative probability of the player needing 70 plays or more to win for the first time. So we need to find the probability for the player to lose in the first 69 tries. This is given by: \[ P(X > 70) = 1 - P(X \leq 69) \] \[ P(X > 70)= 1 - \sum_{k=1}^{69} (1/54)*(53/54)^(k-1) \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Mass Function
The Probability Mass Function (PMF) is a fundamental concept in probability and statistics, particularly relevant in the study of discrete random variables such as the outcome of casino games. It provides the probability that a discrete random variable is exactly equal to some value. In the context of our Wheel of Fortune example, it is used to calculate the probability that a player wins on a specific play of the game.

The formula for the PMF of a geometric distribution is given by
  • \( P(X = k) = p imes q^{(k-1)} \)
  • \( p \) is the probability of success (winning)
  • \( q \) is the probability of failure, which is \( 1 - p \)
  • \( k \) is the number of plays until the first win
The PMF lets us calculate the exact probability that a player wins for the first time on the \( k \)-th play, such as on the 15th play as in our exercise.
Probability of Success
Probability of success refers to the chance of a favorable outcome occurring. In our exercise, this is the probability that a player wins by landing on the joker slot when they spin the wheel. The probability of success is crucial in determining the player's likelihood of winning on a specific attempt.

For the Wheel of Fortune game:
  • The probability of success \( p \) is \( 1/54 \).
  • This value is derived from the fact that only one out of 54 slots is a joker, giving a 1 in 54 chance of landing on it.
Understanding the probability of success allows you to apply it directly in the probability mass function or any other probability calculations related to this game.
Independent Outcomes
Independent outcomes mean that the result of one event does not affect the result of another. In games of chance like spinning a wheel, each spin is an independent event, meaning the outcome of one spin has no influence on the result of the next.

In our Wheel of Fortune context:
  • Each spin (or play) of the wheel is independent of all others.
  • Even if the player wins on one spin, it doesn't change the probability of winning on the next spin; it remains \( 1/54 \).
The concept of independence is important because it ensures that each play is a fresh opportunity to win, unaffected by previous spins. This is a key characteristic that must be satisfied when applying the geometric distribution.
Cumulative Probability
Cumulative probability is used to determine the probability of an event happening at or before a certain number of trials. It helps assess the overall likelihood of an event occurring across multiple attempts.

In our exercise, when dealing with the probability of needing more than a certain number of plays to win, cumulative probability is crucial. For instance:
  • To find the probability of more than 70 plays needed to win for the first time, calculate the probability of failing in the first 69 plays.
  • Using the cumulative probability, \( P(X > 70) = 1 - P(X \leq 69) \), helps determine this likelihood.
  • \( P(X \leq 69) \) is calculated by summing up the probabilities of winning on each play from 1 to 69.
Applying cumulative probability ensures that you consider all possible losing sequences up to the given number of trials, providing a complete view of the likelihood across multiple spins.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A man just bought 4 suits, 8 shirts, and 12 ties. All of these suits, shirts, and ties coordinate with each other. If he is to randomly select one suit, one shirt, and one tie to wear on a certain day, how many different outcomes (selections) are possible?

The probability that a farmer is in debt is 80 . What is the probability that three randomly selected farmers are all in debt? Assume independence of events.

What is meant by the joint probability of two or more events? Give one example.

An investor will randomly select 6 stocks from 20 for an investment. How many total combinations are possible? If the order in which stocks are selected is important, how many permutations will there be?

The probability that a student graduating from Suburban State University has student loans to pay off after graduation is .60. If two students are randomly selected from this university, what is the probability that neither of them has student loans to pay off after graduation?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.