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Playing the slots. Slot machines are now video games, with outcomes determined by random number generators. In the old days, slot machines were like this: you pull the lever to spin three wheels; each wheel has 20 symbols, all equally likely to show when the wheel stops spinning; the three wheels are independent of each other. Suppose that the middle wheel has nine cherries among its 20 symbols, and the left and right wheels have one cherry each. (a) You win the jackpot if all three wheels show cherries. What is the probability of winning the jackpot? (b) There are three ways that the three wheels can show two cherries and one symbol other than a cherry. Find the probability of each of these ways. (c) What is the probability that the wheels stop with exactly two cherries showing among them?

Step by step solution

01

Understand the Scenario

We need to calculate probabilities for a slot machine with three wheels. Each wheel has 20 symbols, and the wheels are independent. Let's consider the probability of landing on cherries as described.

02

Calculate Probability of Jackpot (a)

The probability of winning the jackpot, which means all three wheels show cherries, is calculated by multiplying the probabilities of each wheel stopping on a cherry. For the left and right wheels, this probability is \( \frac{1}{20} \) since each has one cherry. For the middle wheel, it's \( \frac{9}{20} \) because it has nine cherries. The probability is: \[ P(\text{Jackpot}) = \left(\frac{1}{20}\right)^2 \cdot \frac{9}{20} = \frac{9}{8000} \]

03

Identify Scenarios for Two Cherries (b)

There are three scenarios where two cherries appear: 1) cherries on the left and middle wheels, not on the right; 2) cherries on the left and right, not in the middle; and 3) cherries on the middle and right, not on the left.

04

Calculate Probability for Each Two Cherries Scenario

1) **Left and Middle:** Right wheel not cherry \( = \frac{19}{20} \)\[ \frac{1}{20} \times \frac{9}{20} \times \frac{19}{20} = \frac{171}{8000} \] 2) **Left and Right:** Middle wheel not cherry \( = \frac{11}{20} \)\[ \frac{1}{20} \times \frac{11}{20} \times \frac{1}{20} = \frac{11}{8000} \] 3) **Middle and Right:** Left wheel not cherry \( = \frac{19}{20} \)\[ \frac{19}{20} \times \frac{9}{20} \times \frac{1}{20} = \frac{171}{8000} \]

05

Total Probability of Exactly Two Cherries (c)

To find the total probability of exactly two cherries, add the probabilities from each scenario: \[ P(\text{Exactly two cherries}) = \frac{171}{8000} + \frac{11}{8000} + \frac{171}{8000} = \frac{353}{8000} \]

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Key Concepts

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

Random Number Generators

Random number generators (RNGs) are crucial to modern gaming, especially in electronic slot machines. These devices produce sequences of numbers that are unpredictable and simulate randomness. In the context of slot machines, RNGs determine the outcome of each spin.
When you spin the wheels, the RNG ensures that each potential outcome is fair and unbiased. This means that every symbol on a wheel has an equal chance of appearing.
Overall, RNGs play an essential role in ensuring that games are fair and fun, enhancing the user experience by maintaining the element of chance in each play.

Independent Events

The concept of independent events is a key principle in probability. When dealing with probabilities, understanding independence is important for accurate calculations.
Two events are independent if the occurrence or outcome of one event does not affect the occurrence or outcome of another event.
In the case of our slot machine exercise, each wheel operates independently. The result of one wheel does not influence the result of the others. This independence allows us to calculate joint probabilities by simply multiplying the probabilities of individual events. Thus, this concept is pivotal in finding the probability of specific outcomes like winning a jackpot.

Probability of Winning

Calculating the probability of winning in a game of chance involves understanding both the structure of the game and the rules that govern it. For slot machines, the goal might be getting all cherries on the wheels, or perhaps having two cherries.
Let's take a closer look at jackpot probability. To win, all three wheels must display cherries simultaneously. The left and right wheels each show a cherry with a probability of \( \frac{1}{20} \), and the middle wheel has a \( \frac{9}{20} \) probability. Thus, the total probability of a jackpot becomes \( \frac{9}{8000} \).
Understanding the probability helps players grasp their chances of winning and provides insight into the design of the game itself.

Mathematical Symbols

Mathematical symbols are a universal language in probability calculations. They help us represent complex ideas in a concise and understandable form.
In our exercise, fractions such as \( \frac{1}{20} \) are used to denote the probability of specific events. Multiplication of these fractions, like \( \frac{1}{20} \times \frac{9}{20} \), represents the combined probability of independent events.
Finally, the addition of probabilities, such as in determining the probability of exactly two cherries, is performed using symbols: \( \frac{171}{8000} + \frac{11}{8000} + \frac{171}{8000} = \frac{353}{8000} \).
These symbols are essential for efficiently conveying complex mathematical concepts, making it easier for you to calculate and understand probabilities.