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Chapter 6: Problem 99

Roulette Marti decides to keep placing a \(\$ 1\) bet on number 15 in consecutive spins of a roulette wheel until she wins. On any spin, there's a 1 -in- 38 chance that the ball will land in the 15 slot. (a) How many spins do you expect it to take until Marti wins? Justify your answer. (b) Would you be surprised if Marti won in 3 or fewer spins? Compute an appropriate probability to support your answer.

Short Answer

Expert verified
(a) 38 spins on average. (b) Marti winning in 3 or fewer spins has a probability of approximately 7.89%.

Step by step solution

01

Define Probability of Winning Per Spin

Each spin has a probability of winning of \( \frac{1}{38} \), since there is only one winning number (15) out of 38 possible outcomes on a roulette wheel.
02

Calculate Expected Number of Spins Until Win

The expected number of spins to win is the reciprocal of the probability of winning on a single spin. This is because we're dealing with a geometric random variable, and its expected value is \( \frac{1}{p} \). \[\text{Expected spins} = \frac{1}{\frac{1}{38}} = 38\]
03

Determine Probability of Winning in 3 or Fewer Spins

The probability of winning in 3 or fewer spins is the sum of the probabilities of winning on the first, second, and third spin. For a geometric distribution, the probability of winning on the \( n \)-th spin is given by \( (1-p)^{n-1} \cdot p \). Thus, calculate:\[\begin{align*}P(\text{win on 1st spin}) & = p \P(\text{win on 2nd spin}) & = (1-p) \cdot p \P(\text{win on 3rd spin}) & = (1-p)^2 \cdot p\end{align*}\]Combining these,\[P(\text{win in 3 or fewer}) = p + (1-p) \cdot p + (1-p)^2 \cdot p = \sum_{n=1}^{3} (1-p)^{n-1} \cdot p\]Substituting \( p = \frac{1}{38} \):\[P(\text{win in 3 or fewer}) = \frac{1}{38} + \left(\frac{37}{38}\right) \cdot \frac{1}{38} + \left(\frac{37}{38}\right)^2 \cdot \frac{1}{38}\]Compute the result.
04

Compute Exact Probability and Determine Surprise Factor

Calculate the probability from the equation in Step 3:\[P(\text{win in 3 or fewer}) = \frac{1}{38} + \frac{37}{38} \cdot \frac{1}{38} + \left(\frac{37}{38}\right)^2 \cdot \frac{1}{38}\approx 0.0789\]Since the probability is roughly 7.89%, it is relatively low, but not very surprising if it happens within the first three spins.

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Key Concepts

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

Expected Value
Expected value in probability and statistics is a highly useful concept that gives us an average outcome of a random variable over many trials. In the context of roulette, where Marti places a bet each time hoping to win, we want to find out the expected number of bets before she wins.

The situation involves a geometric distribution because it counts the number of trials until the first success. The formula for the expected value of a geometric random variable is straightforward: it is simply the reciprocal of the probability of success, represented as \( \frac{1}{p} \).

In Marti's case, the probability of winning a single spin is \( \frac{1}{38} \). Therefore, the expected value is \( \frac{1}{\frac{1}{38}} = 38 \). This means, on average, Marti can expect to spin the wheel 38 times before securing her win.
Probability
Probability quantifies the likelihood of an event occurring and is central to understanding games of chance like roulette. When Marti bets on number 15, the probability of winning on each spin is \( \frac{1}{38} \).

This probability reflects the long-term frequency of winning if the game were played many times.
  • Calculating probability involves identifying all possible outcomes and determining the fraction that represents successful, or 'favorable', outcomes.
  • For repeated trials, such as multiple roulette spins, probabilities can be summed up for different scenarios to explore questions like, "What's the chance of winning within a certain number of tries?"
This ability to predict outcomes and assess likelihoods makes probability a powerful tool for both theoretical questions and real-world gambling scenarios.
Roulette
Roulette is a popular casino game that revolves around predicting where a ball will land on a spinning wheel. The wheel has numbered slots, and players place bets on numbers they believe the ball might land on.

Each roulette wheel has 38 slots (numbered 1 through 36, plus 0 and 00 in American roulette). The game's allure comes from its simplicity, yet it offers a variety of betting options and outcomes.

Marti's game strategy involves a straightforward bet on the number 15. The excitement and risk come from the low probability of winning on a specific number in any given spin, enticing players to bet and spin repeatedly.
Random Variable
In probability theory, a random variable is a numerical outcome of a random process. Such a variable can take on different values determined by chance, making it foundational to statistical calculations.

There are two main types of random variables:
  • Discrete: These can take on a finite or countable set of values, like the number of spins Marti needs to win.
  • Continuous: These take on an infinite number of possible values within a range.
For Marti's roulette bet, the random variable of interest is the number of spins until she wins. This is modeled by a geometric distribution because it seeks to count trials until the first success (winning), with each trial being an independent event with the same probability of success. Understanding random variables helps in transforming real-life situations into statistical questions that can be analyzed and solved.

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Most popular questions from this chapter

Exercises 27 and 28 refer to the following setting. Choose an American household at random and let the random variable X be the number of cars (including SUVs and light trucks) they own. Here is the probability model if we ignore the few households that own more than 5 cars: Number of cars X: 012345 Probability: 0.09 0.36 0.35 0.13 0.05 0.02 What鈥檚 the expected number of cars in a randomly selected American household? (a) Between 0 and 5 (b) 1.00 (c) 1.75 (d) 1.84 (e) 2.00

Geometric or not? Determine whether each of the following scenarios describes a gcometric setting. If so, define an appropriate gcometric random variable. (a) A popular brand of cereal puts a card with one of five famous NASCAR drivers in each box. There is a 1\(/ 5\) chance that any particular driver's card ends up in any box of cereal. Buy boxes of the cereal until you have all 5 drivers' cards. (b) In a game of \(4-\) Spot Keno, Lola picks 4 \(\mathrm{num}\)bers from 1 to \(80 .\) The casino randomly selects 20 winning numbers from 1 to \(80 .\) Lola wins money if she picks 2 or more of the winning numbers. The probability that this happens is \(0.259 .\) Lola decides to keep playing games of \(4-\) Spot Keno until she wins some money.

Exercises 103 and 104 refer to the following setting. Each entry in a table of random digits like Table D has probability 0.1 of being a \(0,\) and digits are independent of each other. The mean number of 0s in a line 40 digits long is (a) 10. (b) 4. (c) 3.098. (d) 0.4. (e) 0.1.

Smoking and social class \((5.3)\) As the dangers of smoking have become more widely known, clear class differences in smoking have emerged. British government statistics classify adult men by occupation as "managerial and professional" \((43 \% \text { of the }\) population), "intermediate" (34\(\%\) ), or "routine and manual" \((23 \%) .\) A survey finds that 20\(\%\) of men in managerial and professional occupations smoke, 29\(\%\) of the intermediate group smoke, and 38\(\%\) in routine and manual occupations smoke. (a) Use a tree diagram to find the percent of all adult British men who smoke. (b) Find the percent of male smokers who have routine and manual occupations.

Exercises 47 and 48 refer to the following setting. Two independent random variables \(X\) and \(Y\) have the probability distributions, means, and standard deviations shown. X: 125 P(X): 0.2 0.5 0.3 \(\mu_{X}=2.7, \sigma_{X}=1.55\) Y: 2 4 P(Y): 0.7 0.3 \(\mu_{Y}=2.6, \sigma_{Y}=0.917\) Sum Let the random variable \(T=X+Y\) (a) Find all possible values of \(T\) . Compute the probability that \(T\) takes each of these values. Summarize the probability distribution of \(T\) in a table. (b) Show that the mean of \(T\) is equal to \(\mu_{X}+\mu_{Y}\) (c) Confirm that the variance of \(T\) is equal to \(\sigma_{x}^{2}+\) \(\sigma_{\gamma}^{2}\) Show that \(\sigma_{r} \neq \sigma_{X}+\sigma_{Y}\)
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