"The Beauty of Mathematics" Notes

https://cosx.org/2013/01/story-of-normal-distribution-1

One day a buddy, maybe a gambler, asked Di Moffer a question related to gambling: two people A and B gamble in the casino, and the probability of A and B winning is $\mathrm{p\; ,\; q\; =\; 1\; -\; p\; p,\; q=1-p,\; bet \; n\; n\; games.\; The\; two\; agreed:\; if\; the\; number\; of\; games\; A\; wins \; X\; >\; n\; p\; X>\; np,\; then\; A\; pays\; the\; casino \; X\; -\; n\; p\; X-np\; yuan;\; if \; X\; <\; n\; p\; X<np,\; then\; B\; pays\; the\; casino \; n\; p\; -\; X\; np-X\; elements.\; Ask\; the\; casino\; what\; is\; the\; expected\; value\; of\; making\; money.}$

The problem is not complicated, it is essentially a binomial distribution, if  $\mathrm{n\; p\; np\; is\; an\; integer,\; De\; Moffer\; finds\; the\; final\; theoretical\; result\; is}$

$$\mathrm{2npqb(n,p,np)2npqb(n,p,np)}$$

where  $\mathrm{b\; (\; n\; ,\; p\; ,\; i\; )\; =\; (\; n\; i\; )\; p\; i\; q\; n\; -\; i\; b(n,p,i)=(ni)piqn-i\; is\; a\; common\; binomial\; probability.}$