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Chapter 6: Problem 101

A gambler is planning to make a sequence of bets on a roulette wheel. Note that a roulette wheel has 38 numbers, of which 18 are red, 18 are black, and 2 are green. Each time the wheel is spun, each of the 38 numbers is equally likely to occur. The gambler will choose one of the following two sequences. Single-number bet: The gambler will bet \(\$ 5\) on a particular number before each spin. He will win a net amount of \(\$ 175\) if that number comes up and lose \(\$ 5\) otherwise.

Short Answer

Expert verified
Based on the calculation, the expected value of the single-number bet in a game of roulette is \(E = ( -\$5 * \frac{37}{38} ) + (\$175 * \frac{1}{38} )\).

Step by step solution

01

Analyze the given bet

The single-number bet is such that if the gambler's number comes up, they win \$175 after having placed a \$5 bet, hence a net gain of \$175. If any other number comes up, they lose the \$5 placed.
02

Calculate the probability of winning and losing

In a roulette wheel there are 38 possibilities, hence the probability of winning (i.e. the gambler's number coming up) is \(\frac{1}{38}\), and the probability of losing (i.e. any other number coming up) is \(\frac{37}{38}\).
03

Calculate the expected value

The expected value E of a random variable is given by the sum of all values of the variable each multiplied by their respective probabilities. Hence for this situation, the expected value E is calculated as follows: \(E = ( -\$5 * \frac{37}{38} ) + (\$175 * \frac{1}{38} )\).

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Key Concepts

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

Expected Value Calculation
Expected value helps gamblers understand the average outcome of a bet over time. Imagine spinning the roulette wheel repeatedly. The idea is to find what the gambler can expect to win or lose per bet on average.
In mathematical terms, the expected value \(E\) is the sum of all possible outcomes, each multiplied by their probability of occurring.
For a single-number bet, you can calculate the expected value by considering:
  • The probability of losing is \(\frac{37}{38}\). The loss for this outcome is \(-\\(5\).

  • The probability of winning is \(\frac{1}{38}\). The win here is \\)175 (because you earn \\(180 and subtract the initial \\)5 bet).

Putting it together, the formula becomes:\[E = (-\\(5 \times \frac{37}{38}) + (\\)175 \times \frac{1}{38})\]
Solving this gives the expected monetary value per bet. This will show if, on average, the gambler wins or loses money.
Roulette Wheel Probability
Roulette is a popular casino game with a wheel numbered 0 to 37. Understanding probability in this context is crucial for making informed bets.
The wheel consists of:
  • 18 red numbers

  • 18 black numbers

  • 2 green numbers (0 and 00)

Every spin outcome is equally probable because each number is equally likely to appear.
This uniform distribution means that the probability of any specific number (like betting on a single number) appearing is \(\frac{1}{38}\).
Conversely, the probability of any other outcome (not betting number) is \(\frac{37}{38}\).
Knowing these probabilities helps in evaluating the risk and potential reward associated with each bet type.
Single-number Bet Analysis
Betting on a single number in roulette is exciting but carries high risk.
You have to be aware that this type of bet offers a substantial payout if successful, but the chance of winning is low. Specifically, the odds of a single-number winning are \(\frac{1}{38}\), meaning you're more likely to lose each bet.
  • If you win, the payout is significant: you receive \\(175 (considering your initial \\)5 stake).

  • If you lose, you simply forfeit the \$5 bet.

Performing a single-number bet repeatedly will most likely lead to a consistent drain on your finances, given the probabilities involved.
Many enjoy the thrill and potential big win, but knowing the odds and outcomes can guide better betting decisions.

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Most popular questions from this chapter

The Bank of Connecticut issues Visa and MasterCard credit cards. It is estimated that the balances on all Visa credit cards issued by the Bank of Connecticut have a mean of \(\$ 845\) and a standard deviation of \(\$ 270 .\) Assume that the balances on all these Visa cards follow a normal distribution. a. What is the probability that a randomly selected Visa card issued by this bank has a balance between \(\$ 1000\) and \(\$ 1440\) ? b. What percentage of the Visa cards issued by this bank have a balance of \(\$ 730\) or more?

Determine the \(z\) value for each of the following \(x\) values for a normal distribution with \(\mu=16\) and \(\sigma=3\) a. \(x=12\) b. \(x=22\) c. \(x=19\) d. \(x=13\)

The print on the package of Sylvania CFL \(65 \mathrm{~W}\) replacement bulbs that use only \(16 \mathrm{~W}\) claims that these bulbs have an average life of 8000 hours. Assume that the distribution of lives of all such bulbs is normal with a mean of 8000 hours and a standard deviation of 400 hours. Let \(x\) be the life of a randomly selected such light bulb. a. Find \(x\) so that about \(22.5 \%\) of such light bulbs have lives longer than this value. b. Find \(x\) so that about \(63 \%\) of such light bulbs have lives shorter than this value.

Determine the value of \(z\) so that the area under the standard normal curve a. in the right tail is \(.0250\) b. in the left tail is \(.0500\) c. in the left tail is \(.0010\) d. in the right tail is \(.0100\)

How do the width and height of a normal distribution change when its mean remains the same but its standard deviation decreases?
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