Answers:
a) 1/4
b) 1/2
c) 3/11
d) 3/13; the answer from part c is larger
e) 3/4
Each answer is in fraction form
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Work Shown:
Part a)
There are 13 hearts, but we take out 2 cards (2 of hearts, ace of hearts) to get 13-2 = 11 left over.
There are 2 cards removed in each suit (2 card, ace card) totaling 2*4 = 8 cards removed overall. So 52-8 = 44 cards are left over
Divide the two values: 11/44 = 1/4
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Part b)
There are 26 black cards (club, spade, 13 of each). We take out 2 of clubs, ace of clubs, 2 of spades, ace of spades. We're left with 26-4 = 22 black cards.
There are 44 total cards as computed earlier in part a above
22/44 = 1/2
Which effectively says "half of the cards are black"
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Part c)
There are 3 face cards (jack, queen, king) for any one suit. There are 4 suits. So there are 3*4 = 12 face cards.
Divide the number of face cards (12) over the total number of cards after we take out the aces and twos (44). So we have 12/44 = 3/11
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Part d)
If we didn't take out any cards, then we have 12 face cards out of 52 total so 12/52 = 3/13, basically saying 3 out of every 13 cards is a face card.
Let's compare the two fractions 3/11 and 3/13
I'm going to put a double question mark between the two fractions because at this point I don't know which one is larger (or even if they are equal or not)
3/11 ?? 3/13
3*13 ?? 3*11 ... cross multiply
39 ?? 33
That last step has 39 on the left side being larger. So we can replace every double question mark with a "greater than" sign. This is what it would look like after you've done so
3/11 > 3/13
3*13 > 3*11 ... cross multiply
39 > 33
Since the last inequality is true, the first inequality is true. So 3/11 is larger. The probability from part c is larger.
Using a calculator, we can see that
3/11 = 0.2727
3/13 = 0.2308
So this confirms that 3/11 is larger than 3/13.
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Part e)
Subtract the result of part a from 1
1 - (answer from part a) = 1 - (1/4) = 3/4
This rule works because you either get a heart card, or you don't get a heart card. This is the rule of probability complements.
