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A standard deck of cards has 52 total cards divided evenly into 4 suits - there are 13 clubs, 13 diamonds, 13 hearts, and 13 spades.
Ayana and Emil are playing a game that involves drawing 2 cards from a standard deck without replacement to start the game. If neither of the cards are spades, then Ayana goes first. Otherwise, Emil goes first.
Is this a fair way to decide who goes first? Why or why not?
Choose 1 answer:
(A) No, there is a higher probability that Ayana goes first.
B No, there is a higher probability that Emil goes first.
(C) Yes, they both have an equal probability of going first.

A standard deck of cards has 5252 total cards divided evenly into 44 suits - there are 1313 clubs, 1313 diamonds, 1313 hearts, and 1313 spades.\newlineAyana and Emil are playing a game that involves drawing 22 cards from a standard deck without replacement to start the game. If neither of the cards are spades, then Ayana goes first. Otherwise, Emil goes first.\newlineIs this a fair way to decide who goes first? Why or why not?\newlineChoose 11 answer:\newline(A) No, there is a higher probability that Ayana goes first.\newlineB No, there is a higher probability that Emil goes first.\newline(C) Yes, they both have an equal probability of going first.

Full solution

Q. A standard deck of cards has 5252 total cards divided evenly into 44 suits - there are 1313 clubs, 1313 diamonds, 1313 hearts, and 1313 spades.\newlineAyana and Emil are playing a game that involves drawing 22 cards from a standard deck without replacement to start the game. If neither of the cards are spades, then Ayana goes first. Otherwise, Emil goes first.\newlineIs this a fair way to decide who goes first? Why or why not?\newlineChoose 11 answer:\newline(A) No, there is a higher probability that Ayana goes first.\newlineB No, there is a higher probability that Emil goes first.\newline(C) Yes, they both have an equal probability of going first.
  1. Calculate Probability: Calculate the probability that neither of the two cards drawn are spades.\newlineSince there are 3939 non-spade cards in a standard deck of 5252 cards, the probability that the first card drawn is not a spade is 3952\frac{39}{52}.\newlineAfter drawing one non-spade card, there are 3838 non-spade cards left out of 5151 total cards. So, the probability that the second card drawn is also not a spade is 3851\frac{38}{51}.\newlineThe combined probability that both cards are not spades is the product of the two probabilities:\newline(3952)×(3851)(\frac{39}{52}) \times (\frac{38}{51}).
  2. Perform Calculation: Perform the calculation from Step 11.\newline(3952)×(3851)=(34)×(3851)=114204=57102(\frac{39}{52}) \times (\frac{38}{51}) = (\frac{3}{4}) \times (\frac{38}{51}) = \frac{114}{204} = \frac{57}{102}.\newlineThis is the probability that Ayana goes first.
  3. Calculate Probability: Calculate the probability that at least one of the two cards drawn is a spade, which means Emil goes first.\newlineSince the probability that Ayana goes first is 57102\frac{57}{102}, the probability that Emil goes first is 1(57102)1 - \left(\frac{57}{102}\right).
  4. Perform Calculation: Perform the calculation from Step 33.\newline157102=10210257102=451021 - \frac{57}{102} = \frac{102}{102} - \frac{57}{102} = \frac{45}{102}.\newlineThis is the probability that Emil goes first.
  5. Compare Probabilities: Compare the probabilities to determine if the game is fair. If the probabilities were equal, then the game would be fair. However, the probability that Ayana goes first 57102\frac{57}{102} is higher than the probability that Emil goes first 45102\frac{45}{102}.

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