Question

Casino. A casino knows that people play the slot machines in hopes of hitting the jackpot but that most of them lose their dollar. Suppose a certain machine pays out an average of \(\$ 0.92\), with a standard deviation of \(\$ 120 .\) a) Why is the standard deviation so large? b) If you play 5 times, what are the mean and standard deviation of the casino's profit? c) If gamblers play this machine 1000 times in a day, what are the mean and standard deviation of the casino's profit? d) Is the casino likely to be profitable? Explain.

Short answer

The standard deviation is large due to jackpot payouts. For 5 plays, mean profit is $0.40, SD $268.33. For 1000 plays, mean $80, SD $3794.73. Casino is likely profitable due to many plays.

Step-by-step solution

Understanding the Casino's Payout

The standard deviation of the payout is large because the payout structure includes both frequent small losses and infrequent large jackpots. While most losses are small (losing a dollar), winning a jackpot results in a significantly larger payout, which increases the variability.

Finding Mean and Standard Deviation for 5 Plays

For any one play, the expected payout is \(0.92\) dollars, so the expected loss per play is \(1 - 0.92 = 0.08\) dollars. The mean loss for 5 plays is \(5 imes 0.08 = 0.4\) dollars. The standard deviation of the loss is \(\sqrt{5} imes 120 = 268.33\) dollars, due to the independence of each play.

Calculating Mean and Standard Deviation for 1000 Plays

The expected loss per play remains \(0.08\) dollars. For 1000 plays, the mean loss is \(1000 imes 0.08 = 80\) dollars. The standard deviation for 1000 plays is \(\sqrt{1000} imes 120 = 3794.73\) dollars, showcasing the substantial variability when gambling occurs repeatedly.

Analyzing Profitability for Casino

Even with a positive mean profit (from 5 plays or 1000 plays), the standard deviation remains high, indicating high variability. However, the sheer volume of plays (1000 in this case) skews the expected average loss toward profitability over time as the law of large numbers applies.

Key concepts

Standard Deviation

The standard deviation is a crucial concept in understanding how spread out numbers are in a dataset. In the context of the casino slot machines, the standard deviation is a staggering amount of $120. But why is it so high? It's not uncommon in cases where payouts vary widely, like in slot machines. Most slot players will lose a small amount, generally around $1 per play. However, on rare occasions, a player might win a massive jackpot. These large jackpots are infrequent but significantly affect the overall variance in payouts. Thus, you're left with a scenario where the majority see small losses, yet the outlier events - big wins - inflate the standard deviation. Understanding standard deviation helps us see the risk associated with gambling. A higher standard deviation means more variability and unpredictability in outcomes, reminding players to play responsibly.

Expected Value

Expected value is the cornerstone of probability, reflecting the average outcome if an experiment is repeated many times. For the slot machine example, the expected payout for each spin is \(0.92. Hence, the expected loss is calculated as: \[ 1 - 0.92 = 0.08 \] This means, on average, a player loses \)0.08 per play. The expected value helps both players and casinos understand financial expectations over time. For multiple plays, this concept scales up. If you play 5 times, the mean total loss is: \[ 5 \times 0.08 = 0.4 \] This pattern continues with larger numbers of plays. For instance, in 1000 plays, the mean loss is: \[ 1000 \times 0.08 = 80 \] Therefore, the expected value does not change; rather, it multiplies with the number of plays, reinforcing what one might lose on average. Despite the singular expectation of losing, players are often swayed by the small chance of hitting a jackpot.

Law of Large Numbers

The law of large numbers is a fundamental principle in probability and statistics. It states that as the number of trials increases, the average of the results should be closer to the expected value. In the casino setting, it implies that although the outcome of a single slot spin is uncertain, the average payout across a large number of spins will likely align with the expected value. Consider the slot machines: the variability in small numbers of spins is high due to the chance of hitting a jackpot or not. However, over thousands of plays, regardless of big wins or losses, the average loss settles around the mean expectation of $0.08 per spin. In a practical sense for the casino, as players continue to engage with the slot machines, profitability becomes more predictable. By increasing the number of games played, the average loss per player tends to converge on the expected value, ensuring that the casino is likely to profit over time. This principle reassures casinos that despite occasional big payouts, consistent profitability is achievable through volume.