Question

Exercise 4.10 described the game of roulette. Suppose you bet \(\$ 5\) on a single number-say, the number \(18 .\) The payoff on this type of bet is usually 35 to \(1 .\) What is your expected gain?

Short answer

Short answer: The expected gain is approximately -$0.39, which means you can expect to lose around $0.39 per $5 bet on a single number in a game of roulette.

Step-by-step solution

Understand the problem

The problem describes a bet on a single number in a game of roulette. When betting on a single number, there are two possible outcomes: win or lose. If you win, you'll receive 35 times your bet, and if you lose, you'll lose your bet. We need to compute the expected gain, which in this case means finding the weighted average of the possible outcomes. To do this, we'll first find the probability of winning and losing this bet. Then, we'll multiply these probabilities with their respective payoffs. Finally, we'll add the two values together to find the expected gain.

Compute the probability of winning and losing the bet

In roulette, there are 38 numbers on the wheel: 18 red, 18 black, and 2 green (0 and 00). When betting on a single number, the probability of winning is the ratio of the number of successful outcomes over the total number of possible outcomes. Therefore, the probability of winning the bet, P(win) = 1/38 Probability of losing the bet, P(lose) = 37/38

Compute the expected value of the two possible outcomes

The expected gain is calculated as the sum of the product of each outcome's payoff and its probability. Expected gain = (Payoff of winning) * P(win) + (Payoff of losing) * P(lose) If you win, your net gain will be 35 times your bet, minus your initial bet (because you get your bet back as well). In this case, the net gain is 35 * \(5 - \)5 = \(175 - \)5 = $170. If you lose, you will lose your initial bet of $5. Now plugging in the values: Expected gain = (170) * (1/38) + (-5) * (37/38)

Calculate the expected gain

Let's compute the expected gain: Expected gain = (170/38) + (-5 * 37/38) Expected gain = 170/38 - 185/38 Expected gain = -15/38 Expected gain ≈ -$0.39 This means that on average, you would expect to lose around \(0.39 for each \)5 bet on a single number in a game of roulette.

Key concepts

Probability

Probability helps us to understand the likelihood of different outcomes in uncertain situations. In roulette, when you bet on a single number, the chances of winning and losing are determined by probability. To find the probability of winning when betting on one number, you divide the number of winning outcomes by the total number of possible outcomes. In a standard roulette wheel with 38 numbers, betting on one number has a probability of winning as the probability of choosing that specific winning number among the 38. Therefore, the probability of winning is \( \frac{1}{38} \) because there's only one winning scenario out of the total 38 possible outcomes. Conversely, the probability of losing, which includes all other numbers, is \( \frac{37}{38} \) because there are 37 losing outcomes.

Roulette

Roulette is a classic casino game that involves a spinning wheel with numbered pockets. The game of roulette allows players to place bets on either a single number, a range of numbers, or various other criteria like red or black, odd or even numbers. The objective for players who bet on a single number is straightforward: choose one specific number on the wheel before it is spun. If the ball lands on that number, the player wins a payout often at 35 to 1 odds; if not, the player loses their bet. However, the design of the game with its 38 possible outcomes—18 red, 18 black, and 2 green—makes it challenging to win consistently. This setup ensures that the house always has an advantage, making it a popular game for casinos around the world.

Statistical Analysis

Statistical Analysis is essential when evaluating games like roulette to understand expected outcomes over the long term. The expected value lets players know what they might gain or lose on average per bet.For the roulette problem, statistical analysis requires calculating the expected gain, which involves combining probabilities with potential outcomes. It lets players know if a particular bet is profitable or not, in this case using the formula:\[ \text{Expected gain} = (\text{Payoff of winning}) \times P(\text{win}) + (\text{Payoff of losing}) \times P(\text{lose}) \]In the example given, betting \(5 on a single number resulted in an expected value of roughly -\)0.39, indicating a likely loss in the long run. Understanding this expected loss helps in making more informed decisions about whether or not to place such bets.