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Probability is the measure of how likely an event is to occur. In the context of the slot machine game, probability helps us to determine the chances of winning the game by landing on three watermelons. When dealing with slot machines, each wheel has the same number of positions, meaning that each possible outcome has an equal chance of occurring. For our slot machine, each wheel has 11 distinct positions. One of these positions shows a watermelon. Thus, the probability of one wheel landing on the watermelon is \[ \frac{1}{11} \]To find the overall probability of hitting three watermelons (one per wheel), we multiply the probabilities for the individual wheels:\[\left( \frac{1}{11} \right) \times \left( \frac{1}{11} \right) \times \left( \frac{1}{11} \right) = \frac{1}{1331}\]This tells us that the probability of winning the jackpot on this slot machine is very low, which reflects the nature of games of chance.Remember, when calculating the probability of multiple independent events all occurring, you multiply the probability of each event.

Sample space refers to all possible outcomes in a scenario. In our slot machine game, the sample space is all the distinct outcomes for which the wheels could stop. It's important to fully enumerate these possibilities to understand the chances of winning.Each wheel has 11 positions, so a single wheel stopping at any position is one possible outcome. Because the three wheels act independently of each other, you multiply the number of positions on each wheel to find the total sample space:\[11 \times 11 \times 11 = 1331\]The number 1331 represents all the different ways the slot machine wheels can stop. Of those possibilities, only one results in hitting three watermelons simultaneously. This connects directly with the probability of winning, as it gives a foundation for understanding that the event 'three watermelons' is quite rare. Understanding the sample space forms the basis for probability calculations, offering a comprehensive look at the landscape of all possible results.

The payout is the reward you receive if you win the game. For this specific slot machine game, the payout for landing three watermelons is \\(5. This is the enticing element that encourages players to try their luck. However, one must consider both the rarity of winning and the amount spent to play.Let's break the concept into two aspects:

• Winning Payout: In this game, the payout reward is tangible, since you're awarded \\)5 if you succeed in getting three watermelons on a single play.
• Odds against Winning: Given the probability calculated as \(\frac{1}{1331}\), you can imagine that winning isn’t something that happens often. Understanding this rarity affects how often one might expect to receive the payout.

In gambling tools like slot machines, calculating the expected value lets us understand how economically favorable playing the game really is. In this case, the expected value per play is only approximately 0.0038 dollars, which is much less than the cost to play (0.25 dollars per spin). This means you're quite likely to lose more than you gain, typical of such games where the odds favor the house.