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Chapter 15: Problem 46

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

Expert verified
Standard deviation is large due to payout variability. Mean profit is $0.40 (5 plays) and $80 (1000 plays). Casino is likely profitable.

Step by step solution

01

Understanding the Standard Deviation

The standard deviation of \( \$120\)\ indicates large variability in payouts. This occurs because the payouts have a wide range鈥攆rom losing the full dollar to winning a jackpot鈥攚hich means some extreme values skew the distribution.
02

Calculate Mean and SD for 5 Plays

For 5 plays, the mean casino profit is the number of plays times the profit per play: \(5 imes (1 - 0.92) = 0.40\). The standard deviation is the individual standard deviation scaled by the square root of the number of plays: \(\sqrt{5} imes 120 = 268.33\).
03

Calculate Mean and SD for 1000 Plays

For 1000 plays, the mean profit is \(1000 imes (1 - 0.92) = 80\). The standard deviation is \(\sqrt{1000} imes 120 = 3794\).
04

Assessing Profitability

With the mean profit being positive for both 5 and 1000 plays, and considering the law of large numbers, the casino is likely to be profitable in the long run, as increased play will reduce the relative impact of the payout variability due to the standard deviation.

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Key Concepts

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

Understanding Standard Deviation and Variability
Standard deviation is a measure of how spread out the values in a data set are. In the context of a casino slot machine, a standard deviation of \( \\(120 \) indicates a large fluctuation in potential payouts.
This means that the amounts players win can vary widely. To understand why the standard deviation is so large, consider the range of possible outcomes:
  • Players might lose their entire \( \\)1 \) bet, representing a significant negative outcome.
  • Alternatively, they could hit the jackpot, which is a substantially large positive payout.
These extreme values widen the spread of the distribution, leading to a higher standard deviation. This variability can both hurt and benefit a casino's profits depending on the frequency of these extreme payouts.
Mean Calculation and Its Importance
The mean is the average value of payouts for a single play. In this scenario, the mean payout from playing the slot machine is \( \\(0.92 \). Let's break down how we calculate the casino's profit:
  • The mean profit for one play is the difference between what players pay (\( \\)1 \)) and the mean payout (\( \\(0.92 \)). This gives us \( \\)1 - \\(0.92 = \\)0.08 \).
  • For multiple plays, this mean profit is multiplied by the number of plays. For instance, for five plays, it's calculated as \(5 \times 0.08 = \\(0.40 \).
  • For a significant number of plays like 1000, the mean profit is \(1000 \times 0.08 = \\)80 \).
The mean indicates the expected profit over time, helping casinos predict their earnings. It is crucial as it reflects whether a gambling game is profitable.
The Law of Large Numbers in Gambling
The Law of Large Numbers is a pivotal principle. It states that as the number of trials (in this case, plays on a slot machine) increases, the average result will likely get closer to the expected value or mean.Here's how it applies to casino profits:
  • Over a few plays, such as 5, variability can cause significant swings, but the casino still expects \( \\(0.40 \) profit.
  • With 1000 plays, the profit expectation is \( \\)80 \). This is more reliable due to the number of plays stabilizing outcomes closer to the predicted mean.
As more gamblers play this machine, the variable effects of large wins or losses diminish in impact compared to the steady average profit. Essentially, a casino banks on the law of large numbers to ensure profitability over time.
Dealing with Variability in Payouts
Variance in slot machine payouts is both a thrill for players and a risk for casinos. This high variability in outcomes is quantified by the standard deviation, as previously mentioned.When dealing with variability, casinos must recognize:
  • Short-term fluctuations may affect daily profits positively or negatively.
  • Long-term, the expected value or mean should lead to profits, assuming enough plays.
Despite the high standard deviation of \( \$120 \), when gamblers play the machine 1000 times, this variability smooths out, making extreme payouts less impactful. Thus, though exciting, variability poses little threat to long-term profits thanks to larger play volumes ensuring reliance on the average outcomes.

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Most popular questions from this chapter

A golfer keeps track of his score for playing nine holes of golf (half a normal golf round). His mean score is 85 with a standard deviation of \(11 .\) Assuming that the second 9 has the same mean and standard deviation, what is the mean and standard deviation of his total score if he plays a full 18 holes?

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