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Chapter 4: Problem 10

The game of roulette uses a wheel containing 38 pockets. Thirty-six pockets are numbered \(1,2, \ldots, 36,\) and the remaining two are marked 0 and \(00 .\) The wheel is spun, and a pocket is identified as the "winner." Assume that the observance of any one pocket is just as likely as any other. a. Identify the simple events in a single spin of the roulette wheel b. Assign probabilities to the simple events. c. Let \(A\) be the event that you observe either a 0 or a 00\. List the simple events in the event \(A\) and find \(P(A)\) d. Suppose you placed bets on the numbers 1 through 18. What is the probability that one of your numbers is the winner?

Short Answer

Expert verified
Answer: The probability of observing either a 0 or a 00 in a single spin of the roulette wheel is \(\frac{1}{19}\), and the probability of winning if bets are placed on numbers 1 through 18 is \(\frac{9}{19}\).

Step by step solution

01

a. Identify the simple events

In a single spin of the roulette wheel, there are 38 possible outcomes: the ball can land in any of the 36 numbered pockets (1 to 36) or in the two extra pockets (0 and 00). So, the simple events are: {1, 2, ..., 36, 0, 00}.
02

b. Assign probabilities to the simple events

Since there are 38 pockets and each pocket has an equal chance of being the outcome, the probability of each simple event is \(\frac{1}{38}\). Therefore, \(P(1)=P(2)=...=P(36)=P(0)=P(00)=\frac{1}{38}\).
03

c. List the simple events in the event A and find P(A)

Event A is observing either a 0 or a 00. So, the simple events for A are {0, 00}. To find the probability of event A (P(A)), we can add the probabilities of the simple events it contains: \(P(A)=P(0)+P(00)=\frac{1}{38}+\frac{1}{38}=\frac{2}{38}=\frac{1}{19}\).
04

d. Find the probability of winning if bets are placed on numbers 1 through 18

If we place bets on the numbers 1 through 18, we are betting on 18 out of the 38 possible outcomes. To find the probability of winning, we can add the probabilities of each simple event from 1 to 18: \(P(1)+P(2)+...+P(18)=18\times\frac{1}{38}=\frac{18}{38}=\frac{9}{19}\). So, the probability of winning if bets are placed on numbers 1 through 18 is \(\frac{9}{19}\).

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Key Concepts

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

Simple Events Probability
Probability is a way of quantifying the likelihood of various outcomes, and it plays a crucial role in games of chance such as roulette. When we talk about simple events in a probability context, we refer to individual outcomes that are not broken down further. In roulette, each pocket on the wheel is considered a simple event.

In the game of roulette with 38 pockets, each numbered 1 to 36 plus '0' and '00', these pockets represent the simple events. Since the wheel is fair and balanced, we assume that each spin is independent, and every pocket has an equal chance of being the winner. Therefore, the probability of hitting any single number (or simple event) is represented mathematically as \( \frac{1}{38} \). Understanding simple events and their probabilities is foundational for making more complex predictions about the game.
Calculating Roulette Odds
Calculating the odds in roulette involves evaluating the chances that a specific outcome will occur. This process requires accounting for all the possible pockets that a roulette ball could land in after a spin.

To compute these odds, we divide the number of positive outcomes by the total number of possible simple events. For instance, when betting on a single number, the odds are calculated based on there being one favorable outcome and 38 possible pockets.

Calculating Odds for a Single Bet

  • If you're betting on '0', the odds are \( \frac{1}{38} \) since there is only one '0' on the wheel.
  • If you're betting on any single number from 1 to 36, you have a \( \frac{1}{38} \) chance as well.
Identifying and utilizing these odds can help players understand their potential win rate and better strategize their game.
Outcomes of a Roulette Spin
A roulette spin can result in a variety of outcomes depending on where the ball lands on the wheel. The most basic outcome is the ball landing in any of the numbered pockets from 1 to 36, or the 鈥�0鈥� or 鈥�00鈥� pockets unique to American Roulette.

Types of Bets and Their Outcomes

Different types of roulette bets can be categorized by their possible outcomes:
  • Straight-up bet: Betting on a single number. The outcome is just one of the 38 pockets.
  • Split bet: Betting on two adjacent numbers. The outcome is either of the two pockets selected.
  • Street bet: Betting on three consecutive numbers. The outcomes include any of the three pockets in that street.
  • Even/Odd, Red/Black, Half the numbers: You're betting on a larger set of numbers, each with their corresponding pockets as possible outcomes.
Understanding the full range of possible outcomes for a given bet allows players to calculate their odds with greater accuracy and make informed decisions based on the probability of different scenarios.

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Most popular questions from this chapter

Three balls are selected from a box containing 10 balls. The order of selection is not important. How many simple events are in the sample space?

A salesperson figures that the probability of her consummating a sale during the first contact with a client is .4 but improves to .55 on the second contact if the client did not buy during the first contact. Suppose this salesperson makes one and only one callback to any client. If she contacts a client, calculate the probabilities for these events: a. The client will buy. b. The client will not buy.

Suppose \(5 \%\) of all people filing the long income tax form seek deductions that they know are illegal, and an additional \(2 \%\) incorrectly list deductions because they are unfamiliar with income tax regulations. Of the \(5 \%\) who are guilty of cheating, \(80 \%\) will deny knowledge of the error if confronted by an investigator. If the filer of the long form is confronted with an unwarranted deduction and he or she denies the knowledge of the error, what is the probability that he or she is guilty?

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An investor has the option of investing in three of five recommended stocks. Unknown to her, only two will show a substantial profit within the next 5 years. If she selects the three stocks at random (giving every combination of three stocks an equal chance of selection), what is the probability that she selects the two profitable stocks? What is the probability that she selects only one of the two profitable stocks?
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