The Fallen Hearts
When Agnishom asked the prettiest girl in his class to go on a date with him, she pulled out a standard deck of 52 cards.
She said, "You have to choose 2 cards from this pack without replacement. If both the cards belong to the heart suit, we are on."
As she started shuffling the deck, unknown to either of them, 3 cards inadvertently fell out of the pack at random. In effect the deck was consisting of 49 cards when Agnishom chose his two cards.
Given this scenario, the probability that Agnishom goes on the date i.e. that he draws two hearts without replacement can be expressed as b a where a and b are co-prime positive integers.
Find the value of a + b .
The answer is 18.
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1 solution
Discussions for this problem are now closed
An even faster way is observing that the order of selecting cards doesn't matter. Therefore you have to find the probability that the first 2 cards are hearts, which is just 5 2 1 3 ∗ 5 1 1 2 = 1 7 1
Will you please elaborate your method. I can't understand it.
My method is that the order of selecting the cards doesnt matter, i.e. the probability that the 33423rd card is a heart is the same as the probability that the first card is a heart. So the probability of the 4th and 5th card both being hearts is the same as the first two cards being hearts.
Then the probabilty of the first two cards being hearts is just 5 2 1 3 ∗ 5 1 1 2 = 1 7 1
@Siddhartha Srivastava – Thanks for clearing my concepts and a very nice solution
@Prakash Chandra Rai – But my doubt is that if 3 cards are fallen, and at least one of them belongs to hearts, then probability should be affected. Please correct me if I am wrong.
@Prakash Chandra Rai – 3 cards falling is the same as shuffling them to the bottom of the deck. In both cases the process is completely random and the cards can't to be chosen.
@Siddhartha Srivastava – Understood and very very thanks to you.
U r most wlc Agnishom :)
First, we will have to realize that this is just the same thing as choosing 5 random cards with trying to have the last 2 cards heart cards.
To do this, we first notice that the last 2 heart cards can be chosen in 1 3 × 1 2 ways.
We have every freedom in choosing the first three cards except that we cannot choose the two cards we fixed for the choice of the last two. So, we have 5 0 × 4 9 × 4 8 ways to choose the first three cards.
Thus, there are a total of 1 3 × 1 2 × 5 0 × 4 9 × 4 8 ways he can get lucky.
Now, in how many ways can the cards be chosen anyway ? It is obviously ( 5 5 2 ) 5 !
So, P(Agnishom goes to the date) = ( 5 5 2 ) 5 ! 1 3 × 1 2 × 5 0 × 4 9 × 4 8
Thanks for the problem, Satyen! I'll definitely show her :)