# Max of 3 random cards from deck vs max of 3 numbers from 1-13

• I
• member 428835

#### member 428835

Hi PF!

I am wondering the differences between the discrete and continuous case for expected value of minimum of 3 integers uniformly distributed from 1 to 13 vs 3 reals from 1 to 13.

The real case is direct: ##F = ((x-1)/12)^3 \implies f = 3(x-1)/12)^2## for CDF ##F## and PDF ##f##. Thus the expected value for the max of 3 reals in this range is ##\int_1^{13} x f \, dx = 10##. But now for the discrete case: the probability a random variable ##X_i## is less than some integer ##k## I think should be ##P(X_i \geq k) = (13-k+1)/13 \implies P(X \geq k) = ((13-k+1)/13)^3## but I really don't know how to proceed. Is there a direct way to arriving at the CDF that I'm missing?

Why not just ##c(k) = (\frac k {13})^3##? And ##p(k) = (\frac k {13})^3 - (\frac{k-1}{13})^3##?

• member 428835
Just to point out, drawing three cards from a deck is not the same as picking three integers uniformly at random, unless you replace the cards each time you draw.

Also. ##P(X\geq k)## is the cdf, well one minus that is. Were you not sure how to get the pdf (which as Perok points out is just the difference of consecutive pdfs).

Note ##k^3-(k-1)^3## is actually a quadratic polynomial, so the answers are more similar than they might initially appear

• member 428835
I got ##E = \frac{133}{13} \approx 10## for the discrete case.

• member 428835
In general, for a uniform choice from ##1-n## (with replacement) of ##m## cards, then I get the expected value of the highest card to be:
$$E = n - \frac{1}{n^m}\sum_{k =1}^{n-1}k^m$$With ##m = 3##, we have the sum of cubes:
$$\sum_{k =1}^{n-1}k^3 = \frac{n^2(n-1)^2}{4}$$And$$E = n - \frac{(n-1)^2}{4n}$$And with ##n = 13##:
$$E = 13 - \frac{36}{13} = \frac{133}{13}$$

@joshmccraney, your thread title is misleading: "max of 3 random cards from deck vs max of 3 numbers from 1-13"

A deck of cards has 52 cards in it, in four suits. Unless your deck has just 13 cards -- A, 2, 3, ..., J, Q, K -- in one suit, it's different from the set of integers 1 through 13.

The first post seems to be asking a different question -- the difference between a discrete set and a continuous set.

Last edited:
• Klystron, member 428835 and berkeman