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Expected value is a fundamental concept in probability that helps determine the average outcome of a random event over time. When you play the card game described, you have various chances of winning different amounts of money. To decide how much you can expect to win on average whenever you play, you calculate the expected value.

In this game, you can win $50 with a small probability, $25 with a slightly higher probability, and nothing most of the time. Each outcome has a probability associated with it. The expected value is calculated as a weighted average, considering both the amounts you can win and the probabilities of each outcome.

• You win $50 with probability 0.01295.
• You win $25 with probability 0.11765.
• You win nothing with probability 0.8694.

You add up all these values, multiplying each possible win by its probability, to find the total expected winnings. Doing so gives an expected value of approximately $3.59. This tells you that while sometimes you might win big, on average, you make $3.59 per game.

Standard deviation is a statistic that measures the spread or variability of a set of data. In the context of the card game, it helps you understand how widely varied the game outcomes are from the expected value.

When you compute the standard deviation, you begin by calculating the variance, which is the expectation of the squared deviations from the mean. Using the menu of game outcomes and probabilities, variance quantifies how much the results deviate from the expected value.

For this game:

• You first find the expected value of the square of the winnings.
• Subtract the square of the expected value from this to get the variance.
• Lastly, take the square root of that variance to obtain the standard deviation.

For the card game, the standard deviation comes out to be approximately 9.66. This large number relative to the expected value indicates significant variability in your winnings. It implies you could win amounts far from the average $3.59, emphasizing the game's chances of extreme results.

Combinatorics is a branch of mathematics dealing with combinations of objects. It's crucial for calculating probabilities in situations where arrangements and selections matter, like card games.

In the exercise, combinatorics helps determine the probability of the different outcomes by examining the various possible combinations of card draws. Here's how it helps:

• To find the probability of drawing 3 hearts or 3 black cards, you use the combination formula: \( \binom{n}{k} \), which represents the number of ways to choose \( k \) items from \( n \) without replacement.
• For example, \( \binom{13}{3} \) calculates the number of ways to pick 3 hearts from 13, which is crucial for determining associated probabilities.

In essence, combinatorics provides the foundational tool to calculate the likelihood of specific outcomes in the game, offering a precise way to figure out winning chances and build the probability model of your game winnings.

Net profit measures your actual gain or loss after considering costs. In the card game context, you pay to play, so the net profit accounts for the game's entry fee.

The formula for net profit is simple: subtract your cost from your winnings. So, if you win \(50 but paid \)5 to play, your net profit is \(45. Similarly, if you win \)0 after paying \(5, you actually lose \)5.

In this case, the expected value of your winnings per game is \(3.59. When you consider the cost of \)5 to play each time:

• The expected net profit becomes \( 3.59 - 5 = -1.41 \).

This negative expected net profit indicates that on average, you lose $1.41 every time you play the game. In the long run, games with a negative net profit are not financially advantageous, suggesting it's best not to play this game if profit is the goal.