Probability of a deck of cards.

Question

Three cards are randomly selected from a standard deck of 52 cards. What is the probability that the first card is red, the second card is black, and the third card is a heart. You DO NOT REPLACE any drawn cards.

Answer 1

Since the probabilities for the second and third selection depend on the previous selection, they are not independent. You should draw a tree diagram, where each selection has three branches: hearts, red but not hearts (i.e., diamonds), and black (clubs or spades). You don't need to label all 3³ branches, just

hearts-black-hearts and diamonds-black-hearts.

For hearts-black-hearts you have (13/52)(26/51)(12/50).

For diamonds-black-hearts you have (13/52)(26/51)(13/50).

Add those two branches up to get the probability:

(13/52)(26/51)[12/50 + 13/50]=(13/52)(26/51)(25/50)≈0.0637.

Answer 2

The question is asking the total probability (all 3 events combined).  This will require a multiplicative answer of each event.

 

P1=Probability 1st card is red = 0.5   (fairly obvious)

P2=Probability 2nd card is black = 26/51   (recall there is no replacement and ONE red card is gone)

P3=Probability 3rd card is a heart: This one is tricky because we may or may not have drawn a heart on P1

 

Let's examine the sample space for P3.  If things have gone well thus far, we'll have

 

25 red cards (12 hearts, 13 diamonds) OR (13 hearts, 12 diamonds)

25 black cards

 

The probability of drawing a heart is either 12/50 or 13/50 depending on what happened in P1.  Since the probability is equally likeiy, we can average the two:

 

P3=12.5/50

 

Therefore: Total probability=P1*P2*P3= (0.5)(26/51)(12.5/50)=0.0637255 ~6.37%

Answer 3

Hi Jack;

52 cards

26 black

26 red, 13 hearts and 13 diamonds

FIRST CARD IS RED...1/2=26/52

51 cards remain.

SECOND CARD IS BLACK...26/51

50 cards remain.

THIRD CARD IS HEART...

If first card was not heart...13/50

If first card was heart...12/50