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# What is the probability of drawing 3 consecutive hearts in a deck of 52 cards?

I already looked at the answer in my math book, (Algebra 2) I tried doing (13/52)*(2/51)*(1/50). But that lets us arrive at a number that I recall is something to the -7 power which is far from the answer of about 0.01 (Dont remember the answer specificly but it was within 10 times of that. The equation that I used made sense for compound probabilities except for the fact that when geting a first card of Q or K, you cant go consecutive roundabout back into Ace or 2 and can only go consecutive to lesser cards. But If you eliminate THAT, the result will be at an even smaller number! I have done other compound probability probs but this one has me stumped!

Hey! Thanks Guys! I am 99 percent sure that is the answer! That was alot easier than I thought. They should have just said "Three hearts in a row" instead of "Three consecutive hearts" I thought they were asking for permutations with order consecutive numbers ([3 of hearts, 4 of hearts, 5 of hearts], [Jack of Hearts, Queen of Hearts, King of Hearts] ) I'm so stupid! lol :-)

### 15 Answers

- PuzzlingLv 71 decade agoFavorite Answer
You don't have to consider the value of the card at all. Ignore whether the card is a queen, king, or whatever. Just look at the suit.

The probability you draw 3 consecutive hearts is equal to the probability that the first card is a heart and the second card is a heart and the third card is a heart.

What's the chance the first card is a heart? Well there are 13 hearts out of 52 cards.

13/52 = 1/4

What's the chance the second card is a heart? Remember there are only 12 hearts left out of 51 cards:

12/51

What about the third card being a heart? (11 hearts out of 50 cards)

11/50

Multiply this out:

1/4 x 12/51 x 11/50 = 0.0129411765

Answer:

Approx. 1.29%

Edit: Ah! I see your confusion with the word "consecutive". If that were the question, it would be interesting to solve. The ways to pick a set of 3 hearts would be 13 choose 3 = 13 x 12 x 11 / 3! = 286 ways. Assuming that ace was only high or low, there are only 11 ways to form a hand with consecutive hearts (A23, 234, 345, 456, 567, 678, 789, 8910, 910J, 10JQ, JQK). So your answer should be 11/286 times as small.

P(3 hearts that form a consecutive sequence of values)

= 13/52 x 12/51 x 11/50 x 11/286 = 0.000497737557

Approx. 1 in 2000.

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- 1 decade ago
The answer depends on the wording of the question. If you want the probability of 3 CONSECUTIVE hearts (e.g. 7H,8H,9H). Your calculation is correct, first card can be any heart so prob. is 13/52, next card can be one above the first card or one below it, so you get 2/51, then third card has only one option to maintain the consecutive order so the probability is 1/50.

(13/52)*(2/51)*(1/50)= 1.96 *10^-4 = 0.0196 %.

Also I assume that you only draw three cards and don't draw every card looking for a string of three hearts.

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- 1 decade ago
The answer is to the power of -7???? Makes no sense.

If u choose any 3 cards (any suit any number) the probability will be.

1/52 times 1/21 times 1/50 ... Therefor, the probability of 3 consecutive hearts is much higher and equals to

(13/52) * (12/51) * (11/50) = 1716/132600 = 0.0129412

Good luck! :-)

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- notthejakeLv 71 decade ago
you started off right:

(13/52) is the probability of the first heart

now there are 12 hearts left, and 51 cards

(12/51)

and now: 11 hearts and 50 cards (11/50)

so P(3 hearts)= (13/52)(12/51)(11/50) = .0129

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- 1 decade ago
Chance of getting one heart = (13/52)

Chance of pulling second heart = (12/51)

Chance of pulling third heart = (11/50)

Multiply them together and you get 0.01

Why does this work? The top number is the amount of cards available that you want and the bottom number is the amount of total cards.

You start with 13 hearts, take one out and you have 12, take another out and you have 11.

You start with 52 cards, take one out and you have 51, take another one out and you have 50.

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- Anonymous1 decade ago
1.29%

with the first draw you have a 25% chance (13/52)

with the second draw you have a 23.5294% chance (12/51)

with the third draw you have a 22% chance (11/50)

.25*.235294*.22 = .0129412 or 1.29%

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- Its meLv 41 decade ago
(13/52)*(12/51)*(11/50)

your taking away a heart each turn

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- Anonymous1 decade ago
1st draw=1:4

2nd draw=next card hearts is ~12:51 regardless of what previous card was

3rd draw=next card hearts is 11:50 regardless of what previous two cards were.

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