Question

Given a standard deck of 52 cards, 3 cards are dealt without replacement. Using this situation, answer the questions below.


a) What is the Probability that all three cards are queens?


b) Let the first card be the queen of hearts and the second card be the queen of diamonds. Is the probability of drawing the two cards independent? Explain.


c) If the first card is a queen, what is the probability that the second card will not be a queen?


d) If the two cards are queens, what is the probability that you will be dealt three queens
e)if two of the three cards are queen ,which is the probability that the other card is not a queen

Answer 1

Given that 3 cards are dealt without replacement in a standard deck of 52 cards.

Part A:

There are 4 queens in a standard deck of 52 card, thus the probability that the first card is a queen is given by 4 / 52 = 1 / 13.

Since, the first card is not replaced, thus there are 3 queens remaining and 51 ards remaining in total, thus the probability that the second card is a queen is given by 3 / 51 = 1 / 17

Similarly the probability that the third card is a queen is given by 2 / 50 = 1 / 25.

Therefore, the probability that all three cards are queens is given by

$\frac{1}{13} \times \frac{1}{17} \times \frac{1}{25} = \frac{1}{5525}$



Part B:

Yes the probability of drawing a queen of heart is independent of the probability of drawing a queen of diamonds because they are separate cards and drawing one of the cards does not in any way affect the chance of drawing the other card.



Part C:

Given that the first card is a queen, then there are 3 queens remaining out of 51 cards remaining, thus the number of cards that are not queen is 51 - 3 = 48 cards.

Therefore, if the first card is a queen, the probability that the second card will not be a queen is given by 48 / 51 = 16 / 17



Part D:

Given that the first two card are queens, then there are 2 queens remaining out of 50 cards remaining.

Therefore, if two of the three cards are queens ,the probability that you will be dealt three queens is given by 2 / 50 = 1 / 25 = 0.04



Part E:

Given that the first two card are queens, then there are 2 queens remaining out of 50 cards remaining, thus the number of cards that are not queen is 50 - 2 = 48 cards.

Therefore, if two of the three cards are queens ,the probability that the other card is not a queen is given by 48 / 50 = 24 / 25 = 0.96