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# Three cards are chosen at random from a standard 52-card deck. What is the probability that they are not all the same color?

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Three cards are chosen at random from a standard 52-card deck. What is the probability that they are not all the same color?

Dec 31, 2018

#1
+55
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The probability that the three cards are not all red (assuming that there are 26 red cards in the deck and each card is not put back in the deck after it has been drawn) is approximately 0.8824 or 88.24%.

Dec 31, 2018
edited by Cent0rea88  Dec 31, 2018
#2
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Three cards are chosen at random from a standard 52-card deck. What is the probability that they are not all the same suite  (NOT COLOUR)?

Prob that they are all the same suite  is   $$\frac{52}{52}*\frac{\not{12}^4}{\not{51}^{17}} *\frac{11}{50} =\frac{44}{850}=\frac{22}{425}\\$$

Prob that they are not all the same suite = $$1-\frac{22}{425}=\frac{403}{425}\approx94.8%$$

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Dec 31, 2018
edited by Melody  Dec 31, 2018
edited by Melody  Dec 31, 2018
edited by Melody  Dec 31, 2018
edited by Melody  Dec 31, 2018
#3
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Does that mean my answer is wrong then? Shoot. I'll have to se what I did wrong.

Cent0rea88  Dec 31, 2018
#4
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No you might be right.

I found the prob that they are not all of the same suite (not colour)

CPhill is doing it now anyway.

Melody  Dec 31, 2018
edited by Melody  Dec 31, 2018
#5
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Prob that they are not the same color =

1 - prob that they ARE all the same color

P(all the same color)  =

On the first draw, we either get a red or a black  card

The probability  that we match this on the second draw is   25 cards left of the same color / 51 cards left = 25/51

The probability that we match the color on the third draw = 24 cards left of the same color / 50 cards left

So

P( same color)  =   (1) * ( 25/51) * (24/50)  =   25/50 * 24/51  = (1/2) (8/17) =  4/17

So

P (not the same color ) =      1 - 4/17   =   13 / 17

Dec 31, 2018
edited by CPhill  Dec 31, 2018
#6
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(b,b,r) or (r,r,b)

$$p = 2 \cdot \dfrac{\dbinom{26}{2}\dbinom{26}{1}}{\dbinom{52}{3}} = \dfrac{13}{17}$$

Rom  Dec 31, 2018