Suppose the values of the cards are assigned based on their number where possible and for the remaining cards we have $J=11, Q=12, K=13, A=14$. What is the expected value of three drawn cards when I subtract the lowest one?
My (limited) theoretical understanding led me to the following value. Please excuse any notation errors, as I'm an enthusiastic amateur unsure of the proper conventions:
$$\mathbb{E}\left( \text{sum of two highest cards $-$ lowest card} \right)$$ $$ = \frac{1}{13 \choose 3}\sum_{\text{all combinations}} \left( \text{sum of two highest cards $-$ lowest card} \right).$$
I computed this value using Wolfram Alpha and got 15 which doesn't sound senseless to me. However, when I simulated this in Python by drawing three cards 10.000.000 times and computing the value of interest my results suggest the expected value is 14.5. I know there is always randomness involved in simulation but with a sample size like this yielding the same result multiple times I am now convinced my theoretical value was incorrect.
For completeness my Wolfram Alpha code:
{ "input": "totalValue = 0; combinations = Subsets[Range[2, 14], {3}]; Do[sorted = Sort[combination]; totalValue += Total[Take[sorted, -2]] - First[sorted], {combination, combinations}]; N[totalValue/Length[combinations]]" }
As well as my Python code:
np.mean([sum(sorted([rd.randint(2, 14) for _ in range(3)])[1:]) - sorted([rd.randint(2, 14) for _ in range(3)])[0] for _ in range(10000000)])
This is a problem of expectation, and such problems are easier to solve because they deal with averages
We firstly take the numbers to be from $1-13$ (we can always add $1s$ later)
Consider the $13$ points making cuts to divide a line of length $1$ into $14$ equal parts to reflect where the points will fall on a $0-1$ scale, which wil be $1/14, 2/14,...13/14$
And three randomly chosen values will on an average divide the line into $4$ equal parts at $1/4,2/4,3/4$
So to find where the highest sampled point falls, we have
$k_3/14 = 3/4 \Longrightarrow k_3 = 42/4 = 10.5$
Similarly, $k_2/14=2/4 \Longrightarrow k_2 = 28/4 =7$, and $k_1= 14/4 = 3.5$
We now need to add $1$ to each of the above values (to make the numbers $2-14$), thus
$k_3=11.5,\; k_2= 8,\; k_1= 4.5$
and finally, $k_3 + k_2 - k_1 = 11.5+8-4.5 = 15$
which corresponds exactly to the value you got using Wolfram