SOLUTION: two cards are dealt from a pack of ordinary playing cards. find the probability that the second card dealt is a heart?

Question 752622: two cards are dealt from a pack of ordinary playing cards. find the probability that the second card dealt is a heart?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the probability of drawing a red card on the first draw is 13/52 = 1/4.
the probability of not drawing a red card on the first draw is 39/52 = 3/4.

what his means is that 25% of the time you will draw a red card on the first draw and 75% of the time you will not draw a red card on the first draw.

if you draw a red card on the first draw, then the probability of drawing a red card on the second draw is 12/51.

if you do not draw a red card on the first draw, then the probability of drawing a red card on the second draw is 13/51.

25% of the time, the probability of drawing a red card will be 12/51.
75% of the time, the probability of drawing a red card will be 13/51.

total probability is 25% * 12/51 + 75% * 13/51.
evaluate .25 * 12/51 + .75 * 13/51 and you will get 25% probability that you will get a red card on the second draw.

that's the same probability as drawing a red card on the first draw.

if you want to see how this works in real life, assume a much smaller problem.

assume you have 4 cards in the deck.
2 of them are black and 2 of them are white.
what is the probability of drawing a white card on the second draw?

this is a very similar problem.
let's label the 2 black cards b1 and b2.
let's label the 2 white cards w1 and w2.

the deck contains b1 b2 w1 w2

you draw a card out of the deck and then you draw a second card out of the deck.

the possible ways you can draw 2 cards are:

b1 b2
b1 w1
b1 w3
b2 b1
b2 w1
b2 w2
w1 b1
w1 b2
w1 w2
w2 b1
w2 b2
w2 w1
the first column is what you drew on the first draw and the second column is what you draw on the second draw.

on the second draw, you drew 6 black and 6 white.

that's a probability of 50% which is equal to the original probability on the first draw, which is also 50%.

this is because 50% of the time you will draw a black card first and so the probability of drawing a white card on the second draw is 2/3.

this is also because 50% of the time you will draw a white card first and so the probability of drawing a white card on the second draw is 1/3.

your total probability is therefore .5 * 2/3 + .5 * 1/3 which is equal to 1/3 + .5/3 which is equal to 1.5/3 which is equal to 1/2.

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