Part A
We have 20 cards, 4 of which are kings (one of each suit). The probability of getting a king on the first draw is 4/20 = 1/5.
Answer: 1/5
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Part B
We have 20-4 = 16 cards that aren't a king out of 20 total.
The probability of getting a non-king card is 16/20 = 4/5
After selecting the first card and not putting it back, we have 20-1 = 19 cards left. Four of which are a king, so 4/19 represents the probability of getting a king on this draw.
So (4/5)*(4/19) = 16/95 is the probability of getting a king on the second draw where the first draw is not a king.
Answer: 16/95
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Part C
16/20 is the probability of getting no kings on the first draw
15/19 is the probability of getting no kings on the second draw (notice I stepped the values down by 1)
14/18 is the probability of getting no kings on the third draw
Multiplying out the fractions gives
(16/20)*(15/19)*(14/18) = 28/57
I skipped a few steps when multiplying the fractions but you get the idea.
The probability of drawing 3 cards, none of which are a king, is 28/57
Subtract this from 1 to get the complementary probability
1-(28/57) = 57/57 - 28/57 = (57-28)/57 = 29/57
This works because we have two choices: either we get no kings, or we get at least one king. So that's why the two events are complementary and their probabilities add to 1.
Answer: 29/57