S.O.S. Mathematics CyberBoard

Three cards are drawn from an ordinary pack of 52 cards, at random and without replacement. Cards drawn have the following values:

Aces score 1 point,
Tens, Jacks, Queens and Kings score 10 points and
Cards from two to nine score as many points as the numbers they carry (i.e. twos score 2 points, threes score 3 points and so on)

Find the probabilities that
ii) all three cards are of different suits (I found this, it's 197/425)
iii) total score of the three cards is more than 28 given that all three cards are of different suits.

what is the answer to iii)? I'm pretty sure my method is right, but I got 160/2197 and the solution given is 112/2197. If you can get 112/2197, please share. Thanks!

My method:

I found P(scores are 10, 10, 9 and of diff suits) + P(scores are 10, 10, 10 and of diff suits) and divided this by P(cards are of different suits). I got 160/2197 but apparently the answer is 112/2197.

Thanks for your help!

Hello, thesocialnetwork!


I don't agree with any of the answers.
(But that could be my fault.)


(ii) The 3 cards are of different suits.

The first card can be any card: .
. . The second card must be of another suit: .
. . . . The third card must be of yet another suit: .





(iii)

The numerator has two cases:

. .
. . . . There are orders.
. . . . The first card can be any 9: .
. . . . The second is a 10 of another suit: .
. . . . The third is a 10 of yet another suit: .
. . Hence: .

. . . (Of course, the 10's are of different suits!)
. . . . There are choices.
. . . . The first card can be any 10: .
. . . . The second card can be any other 10: .
. . . . The third card can be any other 10: .
. . Hence: .

The numerator is: .

The denominator was determined in part (ii): .


Therefore: .

You didn't show how you get 160/2197, so I can't pin down where your mistake come from. But part (ii) shouldn't give you a prime numerator (the numerator should be divisible by 13^2, as a check).

Counting the ordered triples:
29 points: T-point Suit #1 (4 ways), T-point Suit #2 (4 ways), 9-point Suit #3 (1 way), 3 permutation of card values (T,T,9), 24 ways to choose the suits (#1,#2,#3), in total: 1152 triples.
30 points: T-point Suit #1 (4 ways), T-point Suit #2 (4 ways), T-point Suit #3 (4 ways), 1 permutation of card values (T,T,T), 24 ways to choose suits, in total: 1536 triples.
So there are 2688 triples good triples, out of 52*39*26, i.e. probability 112/2197 as given.

_________________

I noticed my mistake! Many thanks guys.