Random Cards

Pherekydes

gondorff wrote: »

Take 3 cards at random from a deck of 52 cards.

What are the odds that at least one of these three cards is an ace, deuce or a three?


Now I know that if I take one card then the odds are 12/52 obviously. But surely if I take three cards my chances are almost tripled?

In that case, the probability (odds) are 1 less (the probability of not drawing any one of those), which equals

1 - (40/52 * 39/51 * 38/50)

You cannot simply treble the probability of drawing one, as people are inclined to do with probabilities of this nature. To see why, change the question to, "What is the probability of drawing at least one of the five lowest cards in 3 goes?"

The number of cards to choose from becomes 20, and the probability of drawing one in one go is 20/52. 3 times this is 60/52, which is impossible.

MathsManiac

Pherekydes wrote: »

In that case, the probability (odds) are...

At the risk of starting a needless tangent, it's possibly worth pointing out that the "odds" are not the same thing as the probability. The odds of an event E are: P(not E)/P(E).

Pherekydes

MathsManiac wrote: »

At the risk of starting a needless tangent, it's possibly worth pointing out...

Not a needless tangent at all. We all know a lot of people (not mathematicians) use the terms interchangeably.

The probability of getting an ace with one draw of a card is 1/13, while the odds are 12/1, which is P(not E)/P(E).

RoundTower

Adrock-aka wrote: »

The reason I ask is because I got a phone call at 1:45 am last Tuesday morning from my little brother saying he had been shuffling a deck while watching a film for 20 mins or so and dealt himself 4 cards... JJJJ. Aparently he needed to know the odds of that happening THAT VERY MINUTE! And it just couldnt wait until sociable hours.

Even though i had work the next day I was obviously impressed with the feat he had accomplished. Took him years to get that good!!

This is a difficult one, it all depends. Is your brother this guy?

https://www.youtube.com/watch?v=GErpOl3KG_w

gondorff

OK I'm happy with the explanation that Cliste gave. This is good for calculating probability for one hit.
Any fool can calculate the probability for three hits but..

What is the probability of hitting exactly twice?

Do you calculate for

HIT HIT MISS
HIT MISS HIT
MISS HIT HIT

and take an average of the three?

MathsManiac

gondorff wrote: »

OK I'm happy with the explanation that Cliste gave. This is good for calculating probability for one hit.
Any fool can calculate the probability for three hits but..

What is the probability of hitting exactly twice?

Do you calculate for

HIT HIT MISS
HIT MISS HIT
MISS HIT HIT

and take an average of the three?

No. You add them (which only works when they are "mutually exclusive" events - i.e. they don't overlap).

Since all three have the same probability, It's three times any one of them.

gondorff

Please expand. What is the percentage probability of hitting twice?

MathsManiac

The total number of possible outcomes when you choose 3 cards, in a specific order, from a deck of cards is 52*51*50.

Now, how many of these outcomes are you interested in? I'm assuming you're still talking about "hit" meaning getting an ace, two or three.

The total number of ways of getting HIT, HIT, MISS is: 12*11*40
The total number of ways of getting HIT, MISS, HIT is: 12*40*11
The total number of ways of getting MISS, HIT, HIT is: 40*12*11.

So the total number of outcomes of interest is the sum of these three.

When outcomes are equally likely, then the probability is (number of outcomes of interest)/(total number of possible outcomes).

So, the answer is 3(12*11*40) / (52*51*50), which is about 0.12, (i.e., 12%).

MathsManiac

I should also have mentioned, by the way, that since the order the cards are drawn doesn't matter, you could do it with "nCr"s instead:

Total number of (unordered) selections of 3 cards from 52 is 52C3 = 22100.
Number of ways to get 2 of the 12 you want and 1 of the 40 you don't want is (12C2)*(40C1) = 66*40 = 2640.

Probability = (of interest)/total = 2640/22100 = 132/1105 ~= 12%