John draws two cards at random from a pack of 52 cards. After returnin

Question

John draws two cards at random from a pack of 52 cards. After returning them to the pack and shuffling it, George draws two cards at random. The probability that they have exactly one common card is:

A)  \frac{25}{546}

B)  \frac{25}{663}

C)  \frac{48}{663}

D)  \frac{50}{663}

D) None of these

chetan2u · Answer

ways to pick two cards =  52C2=\frac{52*51}{2}=26*51 
let one is common so 2 ways and the second one can be any of the remaining 50 so  2*50

probability =  \frac{2*50}{26*51}=\frac{50}{663}

D

HisHo · Answer

chetan2u 
Can you please check my answer:

We have 2 possibilities:
P of the first card is the common = (1/52)*(50/51)*2 ==> 2 for the sequence
P of the second card is the same.

Adding both: (1/52)*(50/51)*2*2 = 50/663

chetan2u · Answer

Yes  HisHo  you are correct...
p of first card as common = 2/52*50/51..
For second one = 50/52*1/51..
So total = 2*2/52*50/51=50/(51*13)

sandman13 · Answer

Basically, we can mark the cards mentally that were chosen in the first draw. Call them X and Y.

Now we need to choose exactly one out of these two: 2C1 ways
And choose another card from the remaining 50 cards: 50C1 ways
Total ways: 52C2

Prob = 2C1*50C1/52C2 = 50/663.

Answer = D

amplifysat · Answer

why isnt it 51c1, when we pick 1 from 2c1

Bunuel · Answer

It’s not 51C1 because after choosing 1 card from X and Y (2C1), the  other  card must be from the remaining 50 cards - not from 51 (so not Y, and not Y).

Remember - we want  exactly  one common card. So after picking 1 from X and Y, we  must avoid  picking the other marked card.

That’s why it’s 50C1 - not 51C1.

Kinshook · Answer

John draws two cards at random from a pack of 52 cards. After returning them to the pack and shuffling it, George draws two cards at random.

The probability that they have exactly one common card is:

Total ways = 52C2*52C2

Favourable ways = 52C2*2C1*50

The required probability = 100*52C2/(52C2*52C2) = 50/663

IMO D

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