QQuestionAnatomy and Physiology
QuestionAnatomy and Physiology
A game is said to be fair if the expected value (after considering the cost) is 0. This would mean that in the long run, both the player and the "house" or whoever is putting on the game, would expect to win nothing.
If the value is positive, the game is in your favor. If the value is negative, the game is not in your favor. At a carnival, you pay $1 to choose a card from a standard deck. If you choose a red card you double your money, but if you pick a black card you do not get any. (A standard deck of cards has 52 card, 26 of the cards are red.)
1. Complete the probability distribution below.
| Color of card | z | Net Money Won or Lost | P(z) |
| --- | --- | --- | --- |
| Red | | $ | |
| Black | | $ | |
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1. What is the expected value?
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1. Does the player or the carnival have the advantage?
2. Is this a fair game?
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Answer
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Step 1: Complete the probability distribution table.
| Black | -$1 | -$1 | 1/2 |
The probabilities of drawing a red or black card from a standard deck are given by: - P(Red) = number of red cards / total number of cards = 26 / 52 = 1 / 2 - P(Black) = number of black cards / total number of cards = 26 / 52 = 1 / 2 We can now fill in the table as follows: | Color of card | z | Net Money Won or Lost | P(z) | | --- | --- | --- | --- |
Step 2: Calculate the expected value.
E = (1/2) * ($1) + (1/2) * (-$1)
The expected value is calculated as the sum of the product of each outcome's value and its corresponding probability: E = ∑ P(z) * z E = $0
Final Answer
1. The probability distribution table is completed as follows: | Color of card | z | Net Money Won or Lost | P(z) | | --- | --- | --- | --- | | Red | $1 | $1 | 1 / 2 | | Black | -$1 | -$1 | 1 / 2 | 2. The expected value is $0. 3. Neither the player nor the carnival has an advantage in the long run. 4. The game is fair.
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