Question

A roulette wheel contains 38 pocketsthe numbers 1 through \(36,0,\) and \(00 .\) The wheel is spun and the "winning" pocket is recorded, with any one pocket just as likely as any other. Suppose you bet \(\$ 5\) on the number 18 . The payoff on this type of bet is usually \(\$ 35\) for a \(\$ 1\) bet. What is your expected gain?

Short answer

Answer: The expected gain is approximately -\$0.39.

Step-by-step solution

Determine the probability of winning and losing

For a roulette wheel with 38 pockets, the chance of the ball landing in any one pocket is equally likely. Therefore, the probability of winning a bet (i.e., the ball landing on number 18) is: $$ P(\text{winning}) = \frac{1}{38} $$ Conversely, the probability of losing a bet (i.e., the ball not landing on number 18) is: $$ P(\text{losing}) = 1 - P(\text{winning}) = 1 - \frac{1}{38} = \frac{37}{38} $$

Calculate the return on investment for winning and losing

If we win the bet, we get a payoff of \(\$ 35\) for a \(\$ 1\) bet. Since we bet \(\$ 5\), our total winning will be \(\$ 35 \times 5 = \$ 175\). Our net gain in this case will be the total winning minus the amount of our bet, which is: $$ \text{Gain if winning} = \$ 175 - \$ 5 = \$ 170 $$ If we lose the bet, we simply lose the \(\$ 5\) we bet. So, our gain in that case is: $$ \text{Gain if losing} = -\$ 5 $$

Calculate the expected gain

Now that we have the probabilities of winning and losing, as well as the net gain in each case, we can calculate the expected gain by using the following formula: $$ \text{Expected Gain} = P(\text{winning}) \times \text{Gain if winning} + P(\text{losing}) \times \text{Gain if losing} $$ Plugging in the values, we get: $$ \text{Expected Gain} = \frac{1}{38} \times \$ 170 + \frac{37}{38} \times (-\$ 5) $$ $$ \text{Expected Gain} = \$ \frac{170}{38} - \ \$ \frac{185}{38} = - \$ \frac{15}{38} \approx - \$ 0.39 $$ The expected gain of betting \(\$ 5\) on the number 18 is approximately -\$ 0.39. This means that, on average, a player would lose about 39 cents per game when betting on a single number in this scenario.

Key concepts

Roulette Wheel Probability

When it comes to understanding the odds behind casino games, the roulette wheel is a classic example to explore probability. In roulette, each pocket on the wheel has an equal chance of being the 'winning' pocket after a spin. The roulette wheel in our example has 38 pockets, labeled with numbers 1 through 36, along with a 0 and a 00. This implies that the probability of any single number winning is quite low.

Specifically, the chances of our bet on the number 18 winning are calculated as:
\[\begin{equation}P(\text{winning}) = \frac{1}{38}\end{equation}\]
Similarly, the chance that we don't win with our bet on number 18 is:
\[\begin{equation}P(\text{losing}) = \frac{37}{38}\end{equation}\]
This straightforward calculation is crucial for players to understand their odds and sets the stage for deeper insights into expected gains or losses over time.

Expected Value Calculation

The concept of expected value is central in assessing the profitability of bets in gambling. It represents the average amount one can expect to win or lose per bet if the bet were repeatedly placed under the same conditions. To calculate the expected value, we need the probabilities of the different outcomes and the gains or losses corresponding to those outcomes.

The expected gain for betting on a single number in roulette is determined by the formula:
\[\begin{equation}\text{Expected Gain} = P(\text{winning}) \times \text{Gain if winning} + P(\text{losing}) \times \text{Gain if losing}\end{equation}\]
When we insert our values:
\[\begin{equation}\text{Expected Gain} = \frac{1}{38} \times \$170 + \frac{37}{38} \times (\$-5) \end{equation}\]
This calculates to an expected gain of approximately -\$0.39. The negative value indicates a loss, hence, on average, the player tends to lose about 39 cents per game when betting on a single number.

Winning Odds in Gambling

Understanding winning odds is key to gambling and making informed decisions on where to place your bets. In games like roulette, the odds are typically against the player due to the game's structure. A single bet on a number in roulette is a 'straight up' bet, and it offers high payout (35:1 in this case) because the odds of winning are relatively low.

For gambling enthusiasts, recognizing that the house—casino—always has an edge is vital. This edge manifests in the existence of the 0 and 00 pockets that tip the odds in favor of the house. Despite the allure of a big win, the expected gain calculation helps quantify the average outcome, which, more often than not, is a loss for the player. Thus, gamblers must approach these games as entertainment rather than a profit-making endeavor and play responsibly.