Money problem...probability spend in particular time

Question

A child puts money in piggy bank every day , in particular 10 , 20 , 30 , 40 , 50 , or 60 cents with the same probability . Find the probability of spending at least 80 days before having collected 30 euros. Suggestion: use the central limit theorem

Answer 1

Let's choose our unit of money to be euro. Define $X_j=$money deposited on $j$-th day. $\{X_n\}$ are independent and identically distributed. $$\mu=E(X_j)=\sum_{k=1}^6{k\over10}\times{1\over6}={7\over20}\\ \sigma^2=\text{var}(X_j)=\sum_{k=1}^6\left({k\over10}\right)^2\times{1\over6}-\mu^2={7\over240}$$ We'll use Central Limit Theorem to approximate $S=\sqrt{80}({1\over80}\sum_{j=1}^{80}X_j-\mu)$ with $N(0,\sigma^2)$. Required probability $$ P\left({\sqrt{80}}(S+\sqrt{80}\mu)\le30\right)=P\left(S\le{\sqrt5\over10}\right)\approx{1\over\sqrt{2\pi}\sigma}\int_{-\infty}^{\sqrt5\over10}e^{-{x^2\over2\sigma^2}}dx\approx0.9 $$