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

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• member 428835
In summary, the conversation discusses the differences between the discrete and continuous cases for calculating the expected value of the minimum of three integers uniformly distributed from 1 to 13 versus three real numbers from 1 to 13. The real case can be calculated directly using the CDF and PDF, while the discrete case requires a different approach. The conversation also touches on the difference between drawing cards from a deck and picking integers uniformly at random. Finally, the conversation presents a formula for calculating the expected value of the highest card when drawing from a deck of cards and concludes that the expected value for the discrete case is approximately 10.
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

## 1. What is the purpose of comparing the maximum of 3 random cards from a deck to the maximum of 3 numbers from 1-13?

The purpose of this comparison is to analyze the likelihood of obtaining a higher maximum value from a deck of cards versus a set of numbers. This can provide insights into the randomness and distribution of values in a deck of cards.

## 2. How are the 3 random cards selected from the deck?

The 3 random cards are typically selected using a random number generator or by shuffling the deck and then drawing the top 3 cards. This ensures that the selection is truly random and not biased towards certain cards.

## 3. What is the range of possible values for the maximum of 3 random cards from a deck?

The range of possible values for the maximum of 3 random cards from a deck is from 3 to 39. This is because the highest possible value for a card is 13, and 3 cards multiplied together give a maximum value of 39.

## 4. How does the maximum of 3 numbers from 1-13 compare to the maximum of 3 random cards from a deck?

The maximum of 3 numbers from 1-13 will always be 39, which is the same as the highest possible value for the maximum of 3 random cards from a deck. However, the likelihood of obtaining a maximum value of 39 is much lower in a deck of cards due to the randomness of the selection process.

## 5. What other factors may affect the results of this comparison?

Other factors that may affect the results of this comparison include the type of deck being used (e.g. a standard 52-card deck versus a deck with jokers), the shuffling method, and the number of trials conducted. These factors can impact the distribution of values and the likelihood of obtaining a certain maximum value.

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