# Stuck on a few problems

• Sep 6th 2008, 07:14 PM
shootie300
Stuck on a few problems
What is the Probability that a 5-card hand has at least 3 face cards (K,Q,J) ?
.07395 is the answer but how?
12 face cards.
12 choose 1
11 choose 1
10 choose 1
and I've been using 49 choose 1 and 48 choose 1 for the remaining 2 cards, divided by 2598960 (52 choose 5)

What is the probability that a 5 card hand has at least one spade?
.63295 is the answer, didnt know where to start on this one.

13 cards, 4 suits

h, d, c, s, h
• Sep 6th 2008, 07:39 PM
mr fantastic
Quote:

Originally Posted by shootie300
What is the Probability that a 5-card hand has at least 3 face cards (K,Q,J) ?
.07395 is the answer but how?

Mr F says: See main post below for some hints.

12 face cards.
12 choose 1
11 choose 1
10 choose 1
and I've been using 49 choose 1 and 48 choose 1 for the remaining 2 cards, divided by 2598960 (52 choose 5)

What is the probability that a 5 card hand has at least one spade?
.63295 is the answer, didnt know where to start on this one.

Mr F says: 1 - Pr(no spades).

13 cards, 4 suits

h, d, c, s, h

Number of ways you can have at least 3 face cards $= {12 \choose 3} \, {40 \choose 2} + {12 \choose 4} \, {40 \choose 1} + {12 \choose 5} \, {40 \choose 0}$.

Number of ways of choosing 5 cards ${52 \choose 5}$.

Therefore ....
• Sep 6th 2008, 09:29 PM
Shyam
[quote=shootie300;183034]What is the Probability that a 5-card hand has at least 3 face cards (K,Q,J) ?
.07395 is the answer but how?
12 face cards.
12 choose 1
11 choose 1
10 choose 1
and I've been using 49 choose 1 and 48 choose 1 for the remaining 2 cards, divided by 2598960 (52 choose 5)

Number of ways you can choose at least three face cards
= (3 face cards)(2 non-face cards) + (4 face cards)(1 non-face cards) +(5 face cards)(0 non-face cards)

$= C{12 \choose 3} \times C{40 \choose 2} + C{12 \choose 4} \times C{40 \choose 1} + C{12 \choose 5} \times C{40 \choose 0}$

= 171600 + 19800 + 792

= 192192

Number of ways of choosing 5 cards $= C{52 \choose 5}$
= 2598960

Required probability $= \frac {192192}{2598960}$

= 0.7395