Question

All but the "face" cards (Kings, Queens, Jacks) have been removed from a regular deck of 52 playing cards

Draw two cards at random, without replacement. What is the probability that both cards are "Spades"? (Write answer as a fraction reduced to lowest terms) ___________

Based on your answer above, what are the Odds Against both cards being "Spades"? __________

Now draw a third card, again without replacement. What is the probability that this third card is a "Spade" GIVEN that NEITHER of the first two cards drawn are "Spades"? _____________

What would be your answer to the first question asked in this problem if the drawing had been done WITh replacement? ____________

Answer 1

There are 52 cards & 12 face cards in a deck . Now, if we remove 12 face cards, the remaining cards are 40

1. 2 cards are removed in random without replacement from 40 cards. This can be done in ways.

Now, there are 13 spade cards in a deck of 52 cards. When we removed the 3 face cards, there are only 10 cards remaining. Out of these 10 cards, 2 are to selected , which can be done in ways.

Therefore, the probability that both cards are spade =   / = 45 / 780 = 0.058

2. There are 40 cards in total, out of which 10 cards are spade and the remaining 30 cards are non spade.

Now, the Odds Against both cards being "Spades" =    / = 0.5577

3. There are 10 spades & 30 non spades. Now, when 2 cards are already drawn & they both are non spades, we have remaining 28 non spades and 10 spades in a total of 38 cards.

Now, this 1 spade card has to be selected out of 10 in ways.

In case of sample space, 1 card in general can be selected from 38 cards in ways.

Therefore, probability that this third card is a "Spade" GIVEN that NEITHER of the first two cards drawn are "Spades" is given by,

P = / = 0.2631

4. There are 40 cards in a deck out of which 10 are spade.

When we pick the 1st card, the probability of it being spade = 10 / 40

When we pick the 2nd card, the probability of it being spade = 10 / 40

Therefore, their combined probability = 10/40 * 10/40 = 0.0625.

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