In a new card game, you start with a well-shuffled full deck...

In a new card game, you start with a well-shuffled full deck and draw 3 cards without replacement. If you draw 3 hearts, you win $\$ 50$. If you draw 3 black cards, you win $\$ 25$. For any other draws, you win nothing. (a) Create a probability model for the amount you win at this game, and find the expected winnings. Also compute the standard deviation of this distribution. (b) If the game costs $\$ 5$ to play, what would be the expected value and standard deviation of the net profit (or loss)? (Hint: profit $=$ winnings $-$ cost $; X-5$ ) (c) If the game costs $\$ 5$ to play, should you play this game? Explain.

In a new card game, you start with a well-shuffled full deck and draw 3 cards without replacement. If you draw 3 hearts, you win $\$ 50$. If you draw 3 black cards, you win $\$ 25$. For any other draws, you win nothing.
(a) Create a probability model for the amount you win at this game, and find the expected winnings. Also compute the standard deviation of this distribution.
(b) If the game costs $\$ 5$ to play, what would be the expected value and standard deviation of the net profit (or loss)? (Hint: profit $=$ winnings $-$ cost $; X-5$ )
(c) If the game costs $\$ 5$ to play, should you play this game? Explain.

Key Concepts

Probability Model

A probability model assigns probabilities to each possible outcome of a random experiment. In this context, the random variable represents the winnings in a card game, and its probability distribution is defined by the likelihood of drawing specific combinations from the deck. This concept is fundamental for analyzing any game of chance or random process, as it allows us to determine the probability associated with each outcome and thereby assess risk and reward.

Expected Value

The expected value is a measure of the central tendency of a random variable, calculated as a weighted average of all possible outcomes, with the weights being their corresponding probabilities. It provides a single summary statistic for the long-run average outcome of an experiment and is crucial in evaluating the fairness or profitability of a game, investment, or decision.

Standard Deviation

Standard deviation quantifies the amount of variation or dispersion of a set of outcomes in a probability distribution. It is important because it describes how widely the actual outcomes are spread around the mean, offering insight into the risk or volatility involved in a game or any other probabilistic scenario.

Transformation of Random Variables

Transforming random variables, such as by adding or subtracting a constant, alters the distribution of outcomes in a predictable way. In many scenarios, for instance when considering net profit (winnings minus cost), the effect is a shift in the mean by a constant value, while the spread (variance or standard deviation) remains unchanged. This concept is key to understanding how changes in game parameters or costs affect overall success.

Decision Analysis

Decision analysis involves evaluating the expected gains, risks, and uncertainties associated with a choice, such as whether to engage in a game or an investment. By combining the expected value and the risk measures like standard deviation, individuals can make informed decisions that balance potential rewards against possible losses, leading to more rational and optimal choices in the presence of uncertainty.

Transcript

00:01 So we have a game, we have 52 cards in a deck, a well -shuffled deck, and we are taking three cards out without replacement.

00:09 So that is a combination of 52, choose three.

00:13 That's how many different ways we can actually get a card.

00:18 And so that 52, and using my combination button, choose three, there are 22 ,100 ways of having that happen.

00:30 Now, it says if we draw three hearts, now there are 13 hearts in the deck and we want groups of three.

00:39 So 13, choose three is 286 different ways for the hearts.

00:48 And the other is if we draw three black cards, there are 26 black cards and groups of three is 2 ,600.

01:02 So we know that if this happens, then you will win $50.

01:12 We know if this happens, you draw a black card, three black cards, you will win $25.

01:20 And otherwise you lose.

01:22 So it says part a, create a probability model with the amount you will win and find the expected earnings, also find the standard deviation.

01:31 So we have, this is going to be, not including how much it costs.

01:37 So we know you can either earn 50, you can earn 25 or you can earn nothing.

01:43 And the probability is we have 286 ways out of 22 ,100.

01:54 Here we have 2 ,600 ways out of 22 ,100.

02:02 And then if we take the 22 ,100 minus the 286 and the 2 ,600, this is 19 ,214 out of 22 ,100.

02:19 And so we're going to find the mean and the standard deviation for this...

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