Expected value of 3 cards dealt

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Discussion Overview

The discussion revolves around calculating the theoretical expected value of points from a hand of cards dealt from a standard deck, specifically focusing on the expected value of three cards. Participants explore various methods and considerations for determining this value, including the treatment of different card values and combinations.

Discussion Character

  • Mathematical reasoning
  • Exploratory
  • Technical explanation

Main Points Raised

  • One participant seeks to understand how to calculate the theoretical expected value of three cards dealt, expressing frustration with the complexity of the problem.
  • Another participant questions the value assigned to an ace, suggesting it could be 1, 10, 11, or 13, and notes that the expected value might approximate three times the expected value of a single card.
  • A participant provides a detailed breakdown of the combinations for drawing three cards, categorizing them based on the number of tens and non-tens drawn, and suggests a method for calculating the average value.
  • One participant clarifies that an ace is worth 1 point and requests information on the breakdown for four cards instead, indicating a misunderstanding of the original question.
  • A participant references the property of expected values, stating that the expected value of the sum of two random variables equals the sum of their expected values, which could simplify the calculation.
  • Another participant calculates the expected value of one card as 85/13, assuming specific values for the cards, and concludes that the expected value for three cards can be derived from this single card value.

Areas of Agreement / Disagreement

Participants express differing views on the value of the ace and how to approach the calculation of expected values. There is no consensus on the method for calculating the expected value of three cards, as various approaches and assumptions are presented.

Contextual Notes

Participants have not reached a definitive agreement on the value of the ace or the method for calculating the expected value of multiple cards. The discussion includes various assumptions and breakdowns that may depend on specific interpretations of card values.

froggy21
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If each card on a regular deck has points that corresponds to their number (like 2 of hearts is 2 points, 7 of clubs is 7 points), the Jack, Queen, King each being 10 points...what's the expected value of your opponent's hand if you deal them 3 cards?

I know the empirical expected value...but I'd like to know -how- to get the theoretical expected value, please : )

Help please D: I've been stewing over this question for days now. The only way I can think of doing this is by doing a tree diagram to get each probability but that'll have like 1000 end branches -headdesk-
 
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What's an ace worth? 1, 10, 11, 13?

The expected value wouldn't be too far off from three times the expected value of one card.
 
I'll assume the ace is valued at something other than 10.

So there are 52 * 51 * 50 ways to choose three cards from a deck. Of those, the breakdown is

216 ways to draw 3 of the same non-10
10368 ways to draw 2 of the same and one different, none 10
32256 ways to draw 3 different non-10s
5184 ways to draw a 10 and 2 of the same non-10s
55296 ways to draw a 10 and 2 different non-10s
25920 ways to draw 2 10s and a non-10
3360 ways to draw 3 10s

So calculate the average value for each, multiply, add, and divide.
 
Ah, yes, Ace is worth 1 points : )

Uhm, sorry, would you happen to know the breakdown for 4 cards? I misread the question and apparently it's the expected value of 4 cards dealt. I tried doing the breakdown myself but I always seem a few hundred thousand short of the total ways.

Thank you very much for all the help!
 
Note that E(X+Y)=E(X)+E(Y) holds regardless of the dependence between X and Y - so you won't need to work out all 13^4 combinations.
 
To expand on bpet's remark, the expected value of one card is 85/13, assuming an ace is 1 and Jack, Queen, King are 10 each. Let's say the value of the ith card is [tex]X_i[/tex]. Then the expected value of 3 cards is

[tex]E(X_1 + X_2 + X_3) = E(X_1) + E(X_2) + E(X_3) = 3 \times 85/13[/tex].

That's all it takes.
 

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