In the “quiz show” example in the notes (§1.2.6 but reproduced alsobelow).
Setting:$\;$ In a quiz show a random integer $r$ between $1$ and $10$ is generated. Another, independent such random number s will then be generated, but before that happens, you are invited to guess whether $s$ will be greater than or less than $r$. If you are correct, then you win s pounds. If you lose (or if $s = r$) then you win nothing.
Questions:
- At what value of $r$ does it change from being correct to bet on s being less than $r$ rather than greater?
- Show that if the range is $1, \dots ,N$ (instead of $1, \dots , 10$) then in the limit as $N$ tends to infinity the change happens at about $\frac{N}{k}$ where $k$ is a constant that you should evaluate
Attempt:$\;$ I understand the answer to 1. is $r = 8$; however, I cannot figure out how to deduce $k$ at all.
One way to approach this is to imagine playing $N$ times with the same $r$ and $s$ taking all the values from $1$ to $N$. You want to choose your bet to win the maximum total. This gives you the expected win times $N$. In the case $N=10$, betting larger on $8$ wins $19$ while betting smaller on $8$ wins $28$, so you should bet smaller on $8$. Betting larger on $7$ wins $27$ while betting smaller on $7$ wins $21$ so you should bet larger on $7$.
For the second part, if you bet smaller on $\frac Nk$ you win $\frac 12 \frac Nk(\frac Nk-1)$ assuming $\frac Nk$ is whole. If you bet larger you win $\frac 12(N-\frac Nk)(N+\frac Nk+1)$. We want to set these equal, so $$\frac 12 \frac Nk(\frac Nk-1)=\frac 12(N-\frac Nk)(N+\frac Nk+1) \\ \frac {N^2}{k^2}-\frac Nk=N^2-\frac {N^2}{k^2}+N \\2\frac {N^2}{k^2}=N^2+N+\frac Nk$$ For large $N$ the terms in $N^2$ will dominate so we can ignore the terms in $N$ and we see $k=\sqrt 2$. Note that $\frac {10}{\sqrt 2}$ is very close to $7$