Question

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A hand of 13 cards is dealt from a standard deck of 52 playing cards. What...

A hand of 13 cards is dealt from a standard deck of 52 playing cards. What is the probability that it contains more spades (♠) than hearts (♡) given that the hand contains at least two spades?

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Expert Solution

Answer:

Given that 13 cards drawn from deck of 52.

Here condition is spades count shouldbe more than hearts count in selection of 13.

So Let us consider Spades(S) = 13 , Hearts (H) = 13 and Remaining(R) = 26.

Number combinations to get 13 cards from 52 id (52 c 13) = 635013559600

Here Possibilities are

If R=0 in this 13,

7 Spades & 6 heart

8 Spades & 5 heart

9 Spades & 4 heart

10 Spades & 3 heart

11 Spades & 2 heart

12 Spades & 1 heart

13 Spades & 0 heart

We have 7 chances.

If R=1, so S + H = 12 and S > H.

7 Spades & 5 heart

8 Spades & 4 heart

9 Spades & 3 heart

10 Spades & 2 heart

11 Spades & 1 heart

12 Spades & 0 heart

We have 6 chances.

If R=2, so S + H = 11 and S > H.

6 Spades & 5 heart

7 Spades & 4 heart

8 Spades & 3 heart

9 Spades & 2 heart

10 Spades & 1 heart

11 spades & 0 heart

we have 6 chances

If R = 3, so S + H = 10 and S > H.

6 Spades & 4 heart

7 Spades & 3 heart

8 Spades & 2 heart

9 Spades & 1 heart

10 spades & 0 heart

wE HAVE 5 chances

If R =4 , So S+ H= 9 and S > H.

5 Spades & 4 heart

6 Spades & 3 heart

7 Spades & 2 heart

8 Spades & 1 heart

9 spades & 0 heart

wE HAVE 5 chances

If R = 5, So S+ H= 8 and S > H.

5 Spades & 3 heart

6 Spades & 2 heart

7 Spades & 1 heart

8 spades & 0 heart

wE HAVE 4 chances

If R = 6, Then S + H =7 and S > H.

4 Spades & 3 heart

5 Spades & 2 heart

6 Spades & 1 heart

7 spades & 0 heart

wE HAVE 4 chances

If R = 7, then S + H = 6 and S > H.

4 Spades & 2 heart

5 Spades & 1 heart

6 spades & 0 heart

wE HAVE 3 chances

If R = 8, then S + H= 5 and S > H.

3 Spades & 2 heart

4 Spades & 1 heart

5 spades & 0 heart

wE HAVE 3 chances

If R = 9, then S + H= 4 and S > H.

3 Spades & 1 heart

4 Spades & 0 heart

wE HAVE 2 chances

If R = 10, then S + H= 3 and S > H.

2 Spades & 1 heart

3 Spades & 0 heart

wE HAVE 2 chances

If R = 11 then S + H = 2 and S > H.

2 Spades & 0 heart

wE HAVE 1 chances

If R = 12 then S + H = 1 and S > H.

1 Spades & 0 heart

wE HAVE 1 chances

If R = 13 then S + H = 0 and S > H.

No chance

wE HAVE 0 chances

So total selection P(R=0)+ P(R=1) +.....+ P(R=13)

= 7 +6+6+5+5+4+4+3+3+2+2+1+1+0

= 49

P(X) = 49/635013559600

milcah answered 3 months ago

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