Can someone explain this math probability question to me.?

Question

Two cards are selected at random from a standard 52-card deck without replacement. Find P(first is a diamond and the second is not a face card).

Answer 1

Since order matters, there are
52*51 possible two-card hands.

Out of these, we count the number of hands satisfying the requirements. There are two cases:
1) The first card is the jack, queen or king of diamonds. Then, there are 40 non-face cards left to choose for the second card. Thus, there are 3*40 = 120 of these.

2) The first card is some other diamond. Then, there are 39 non-face cards left to choose for the second card. Thus, there are 10*39 = 390 of these.

So there are 120 + 390 = 510 possible hands satisfying the requirements.

The probability is then 510 / (52*51) = 10/52 = 5/26

****************************** ******************************
If you don't like the counting method, you can work directly with probabilities:

3/52 of the time we will draw a face card diamond first, and then the probability of drawing a non-face card second is 40/51.

10/52 of the time we will draw a non-face card diamond first, and then the probability of drawing a non-face card second is 39/51. (we used one of the forty available non-face cards for the first card)

The total probability is then:
(3/52)*(40/51) + (10/52)*(39/51)
= 120/(52*51) + 390/(52*51) = 510/(52*51)
= 10/52 = 5/26

****************************** ******************************
note to Katherine:
you won't always have a 40/51 probability of drawing a non-face card second. if the first card drawn wasn't a face card, the probability would be 39/51.


note to oregfiu:
There are only 3 face cards per suit (J, Q, K), not 4.
The technique is sound however and parallels the probability method I described above.

Answer 2

well first you know that you would have a 1/4th chance of getting a diamond because there are four suits
then the card would be not a face card so you subtract 11 from 51 because there could be a possible 11 face cards in the deck
so you would have a 40/51 chance of pulling a non face card
then you multiply the two chances 1/4 x 40/51 = 40/204
that simplified would be 10/51

Answer 3

P (diamond, non face) = P (face diamond, non face) + P (non face diamond, non face) =
= (4/52) * (40/51) + (9/52) * (39/51) = 511 / (51*52) = 0,192685 = 19.2685 % probability

========================

to Zaphod:

Thanks. I was not sure about aces. In my set
http://s210.photobucket.com/albums/bb64/oregfiu/cards/?action=view¤t=cards.jpg
all cards are facecards :-)