---
# System prepended metadata

title: Lab Expectations (#27)

---

# Lab Expectations (#27)

## Problem: Expectations
### a.
total number of poker hands:
n=52 k=5
\\[ {52 \choose 5} = \frac{(52)_5}{5!}. = 2,598,960\\]

single pair:
${13 \choose 1}*{4 \choose 2}*{12 \choose 3}*{4 \choose 1}*{4 \choose 1}*{4 \choose 1} = 1,098,240$

full house:
${13 \choose 1}*{4 \choose 3}*{12 \choose 1}*{4 \choose 2} = 3,744$

flush (not royal or straight):
${4 \choose 1}*({13 \choose 5}-10) = 5108$

assuming $1 for only single pair $10 for a full house and $100 for a non-royal/straight flush

$1(1,098,240/2,598,960) + $10(3,744/2,598,960) + $100(5108/2,598,960) - $3((1,098,240+3,744+5108)/2,598,960)

-$0.64

### b.
1d20$\geq$12 => (20-11)/20 = 9/20


|dice1|dice2|total|
|-----|-----|-----|
|1    |1    |2    |
|1    |2    |3    |
|1    |3    |4    |
|1    |4    |5    |
|2    |1    |3    |
|2    |2    |4    |
|2    |3    |5    |
|2    |4    |6    |
|3    |1    |4    |
|3    |2    |5    |
|3    |3    |6    |
|3    |4    |7    |
|4    |1    |5    |
|4    |2    |6    |
|4    |3    |7    |
|4    |4    |8    |

2(1/16)+3(2/16)+4(3/16)+5(4/16)+6(3/16)+7(2/16)+8(1/16) = 5

5(9/20) = 2.25 damage on average
