You decide to offer a gambling game with cards to your...

You decide to offer a gambling game with cards to your friend, Amos. You mark each card with a number from 1 to 52 ; that is, you write 1 on the first card in the deck, 2 on the second card, etc., to 52 on the last card. You shuffle the cards. If the top four cards are in ascending order you pay Amos $$\$ 20$$; otherwise he pays you a dollar. (By ascending order we mean, for example, the top card is 7 , the second card 12 , the third card is 40 , and the fourth card is 47. .) What is the probability that Amos will win? What is your average winning per play? [Hint: In how many orders can the top 4 cards be arranged?]

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Key Concepts

Expected Value

Expected value is a fundamental concept in probability and statistics that represents the average outcome over a long series of trials. It is calculated by multiplying each possible outcome by its corresponding probability and summing these products, providing a measure of the long-term gain or loss in games of chance or other probabilistic processes.

Probability Calculation

This key concept involves computing the likelihood of an event by taking the ratio of the number of favorable outcomes to the total number of possible outcomes. Understanding how to accurately count both these figures is fundamental in determining precise probabilities in any randomized scenario.

Combinatorial Counting

This concept involves determining the total number of outcomes in a scenario by using techniques such as permutations and combinations. It is essential for understanding how many ways items can be arranged or selected, especially when order matters, which is key in calculating probabilities.

Permutations

Permutations refer to the arrangement of all or part of a set of objects where the order is significant. The total number of permutations of a set gives insight into how many possible sequences can be formed, which is critical when the event of interest depends on the order of elements.

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