Card Hand probabilities: Discrete Math

SammC

1. Homework Statement
In the following you are given a 5-card hand from a 52 card deck.
a) given that you have at least one ace, what is the probability you have at least 2 aces?
b) given that you have the ace of diamonds, what is the probability that you have another ace?
c) given that you have a red ace, what is the probability you have another ace?
d) Suppose you select a card from your hand at random and it is an ace. What is the probability you have another ace.

2. Homework Equations
pr(not a) = 1 - pr(a)
pr(A given B) = Pr(A and B) / Pr(B)

3. The Attempt at a Solution

C(X,Y) is X choose Y here.

Pr(A): The chance of drawing AT LEAST one Ace is 1 - C(48,5)/C(52,5) =~ .6588

Pr(B): The probability of having at least 2 aces: ?
Pr(A and B): ?

pr(C): That you have the ace of diamonds in a 5 card hand is 1-C(51,5)/C(52,5) =~ .096
Pr(C and B):?

pr(D): That you have at least one red ace in a 5 card hand: 1-C(50,5)/C(52,5) =~ .1848
(im not sure if i want the probability of having exactly 1 red ace, or at least one)
Pr(D and B):?

Finding all of the "?" probabilities should allow me to calculate problems a-c, but i really dont know how to find PR(X AND Y) with these problems, nor do I know how to get the probability of having at least two aces.

Related Engineering and Comp Sci Homework Help News on Phys.org

jambaugh

Gold Member
Probability of having at least two aces...
Hmmm...
You can calculate the probability of having exactly 4 aces + exactly 3 aces + exactly 2 aces.

Pr(4) = C(4,4)C(48,1)/C(52,5)
Pr(3) = C(4,3)C(48,2)/C(52,5)
Pr(2) = C(4,2)C(48,3)/C(52,5)

SammC

Thanks, that makes sense.

How do you go about getting the probability of the union though?

jambaugh

Gold Member
Since you are enumerating distinct hands (no hand with exactly 2 aces is a hand with exactly one ace) you can simply add the probabilities.
Pr(x>1)= Pr(x=2) + Pr(x=3) + Pr(x=4).

SammC

what about pr(1 ace red AND at least 2 aces) one is not necessarily in the other in this case, same with 1 ace diamond

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving