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Chapter 4: Problem 100

A slot machine has three slots; each will show a cherry, a lemon, a star, or a bar when spun. The player wins if all three slots show the same three items. If each of the four items is equally likely to appear on a given spin, what is your probability of winning?

Short Answer

Expert verified
Answer: The probability of winning on the slot machine is 1/16 or approximately 0.0625.

Step by step solution

01

Determine the total outcomes

Since each slot can show 1 of the 4 items, there are a total of 4x4x4=64 possible outcomes for the three slots after spinning.
02

Identifying winning outcomes

We win only when all 3 slots show the same item. It could be either cherry, lemon, star, or bar. So, there are 4 possible winning outcomes.
03

Calculate the probability

The probability of winning is the ratio of winning outcomes to the total outcomes. So, to find the probability, we can use the formula: P(winning) = \frac{winning \thinspace outcomes}{total \thinspace outcomes} = \frac{4}{64}
04

Simplify the probability

Now, we can simplify the probability by reducing the common factors in the ratio, which gives: P(winning) = \frac{1}{16} Thus, the probability of winning on the slot machine is 1/16 or approximately 0.0625.

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Key Concepts

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

Slot Machine
A slot machine is an exciting game of chance that you often find in casinos. It features three spinning slots, each capable of displaying different symbols like cherries, lemons, stars, or bars when stopped.

When you pull the lever or press the button, the slots revolve and stop randomly, giving you a combination of these symbols.
  • The thrill of a slot machine is largely due to its unpredictability.
  • Your main objective is to get all slots to show the same symbol for a win.
  • Although the process seems random, each symbol has a calculated probability of appearing.
Playing a slot machine can be fun, but it's important to understand the underlying probabilities to gauge your chances of winning.
Outcomes
In probability, the term "outcomes" refers to the possible results that can occur from a random event. For the slot machine, each spin gives a variety of outcomes depending on the combination of symbols.

Each slot has four different possible symbols after a spin, meaning the total number of possible outcomes for all three slots is calculated by multiplying these possibilities together, yielding \[4 \times 4 \times 4 = 64.\]
  • This number, 64, represents all the conceivable outcomes after spinning the slot machine once.
  • Only one specific type of outcome, where all slots show the same symbol, results in a win.
  • Considering these outcomes helps in understanding why the probability of winning is calculated as it is.
Recognizing the possible outcomes is crucial in grasping the structure of probability problems involving slot machines and other related games of chance.
Winning Probability
The winning probability is a measure of how likely you are to achieve the desired result when playing a game of chance, like a slot machine. In this context, the result we're aiming for is having all three slots display the same symbol.

To determine this probability, consider the number of favorable outcomes, which is the scenario where all slots show the same symbol.
  • In this case, that number is 4, as there are four different symbols.
  • The probability of winning is therefore \[\frac{4}{64}.\] calculated as the ratio of successful outcomes to the total outcomes.
  • Understanding this concept helps reinforce the logic behind the calculations and how probability shapes our expectations when engaging with random events.
Probability gives you a quantifiable understanding of how often you can expect to hit the jackpot when playing the slot machine. It helps set realistic expectations and better decision-making.
Simplification of Probability
Simplifying probability is crucial as it makes results more understandable and manageable. In the slot machine example, we started with the fraction \(\frac{4}{64}\).

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which in this case is 4.
  • Upon simplification, \[\frac{4}{64}\] simplifies to \[\frac{1}{16}.\]
  • This gives us a clearer view of the likelihood, converting it to a value that is simpler to interpret.
  • Simplification can apply to any probability fraction, allowing for easier comparisons with other scenarios or probabilities.
The act of reducing fractions is an essential mathematical exercise that enhances comprehension of results. This understanding helps in not just gaming strategies but in various real-world applications where probability plays a significant role.

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Most popular questions from this chapter

A random variable \(x\) has this probability distribution: $$\begin{array}{l|llllll}x & 0 & 1 & 2 & 3 & 4 & 5 \\\\\hline p(x) & .1 & .3 & .4 & .1 & ? & .05\end{array}$$ a. Find \(p(4)\). b. Construct a probability histogram to describe \(p(x)\). c. Find \(\mu, \sigma^{2},\) and \(\sigma\). d. Locate the interval \(\mu \pm 2 \sigma\) on the \(x\) -axis of the histogram. What is the probability that \(x\) will fall into this interval? e. If you were to select a very large number of values of \(x\) from the population, would most fall into the interval \(\mu \pm 2 \sigma\) ? Explain.

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A food company plans to conduct an experiment to compare its brand of tea with that of two competitors. A single person is hired to taste and rank each of three brands of tea, which are unmarked except for identifying symbols \(A, B\), and \(C\). a. Define the experiment. b. List the simple events in \(S\). c. If the taster has no ability to distinguish a difference in taste among teas, what is the probability that the taster will rank tea type \(A\) as the most desirable? As the least desirable?

Professional basketball is now a reality for women basketball players in the United States. There are two conferences in the WNBA, each with six teams, as shown in the table below. \(^{3}\) Two teams, one from each conference, are randomly selected to play an exhibition game. a. How many pairs of teams can be chosen? b. What is the probability that the two teams are Los Angeles and New York? c. What is the probability that the Western Conference team is not from California?
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