You are dealt two playing cards from a standard deck (without replacement). What is the probability that one of the cards is the 4 of diamonds?

Question

parthkohli · Answer

I $\heartsuit $ probability.

unklerhaukus · Answer

$$\boxed{\color{red}{4\diamond}}$$

parthkohli · Answer

P(the first card is four of diamonds) = 1/52.
P(the second card is four of diamonds) = 51/52 * 1/51

Add both.

unklerhaukus · Answer

I get 2/52

parthkohli · Answer

Good.

unklerhaukus · Answer

|dw:1451392732510:dw|

parthkohli · Answer

yes, I think you can think of it like this. Is 1/26 right though?

ganeshie8 · Answer

unklerhaukus · Answer

This was my first thought,

unklerhaukus · Answer

1-51/52*50/51 = 1 - 50/52 = 2/52

unklerhaukus · Answer

hmm ok, i guess it really was that easy

unklerhaukus · Answer

i've been studying too much QM, no quantum interference here

unklerhaukus · Answer

If instead there is replacement, the probability is greater right?

1/52 + 51/52 * 1/52 + 1/52*1/52  =  2/52 + 1/52^2  > 2/52

ganeshie8 · Answer

ganeshie8 · Answer

P("at least one time 4diamond") = 1 - P("no time 4diamond")

unklerhaukus · Answer

but 
1/26 = 0.03846 > 0.03809

ganeshie8 · Answer

Now that looks very interesting...

unklerhaukus · Answer

what formulae are we using here?

parthkohli · Answer

The probability of getting it with replacement should obviously be lower, no?

unklerhaukus · Answer

hmm, it's not so obvious

unklerhaukus · Answer

Why is is less?

parthkohli · Answer

One more card in the deck makes it harder to get the intended one.

ganeshie8 · Answer

what if the replacement card itself is the intended one ?

parthkohli · Answer

but that is only one case compared to the fifty-one others.

ikram002p · Answer

1/52+1/51

unklerhaukus · Answer

The chance of getting 4 of diamonds twice, is relatively insignificant.

ganeshie8 · Answer

`If instead there is replacement, the probability is greater right?`

1/52*51/52 + 51/52*1/52 + 1/52*1/52  =  (2*51+1)/52^2

unklerhaukus · Answer

ah  @ganeshie8, that's it

and the previous case  was 
 1/52 + 1/52 = 2*52/52^2

comparing numerators 
2*52 > 2*51+1

parthkohli · Answer

oh so was I wrong

unklerhaukus · Answer

I understand it better now, thanks to all

unklerhaukus · Answer

you were not wrong at all @ParthKohli

unklerhaukus · Answer

1/26 =  2*52/52^2

parthkohli · Answer

yes, got it