Could someone check my solution and tell me what am I doing wrong in the second probability?
Thank you.
A dice is tossed 1000 times. Lex $x$ be the number of times of getting 4 points. Compute $P(x\ge170)$ and $P(162\le x\le 171)$.
Solution.
As $x\ge30,$ $x$ can be approximated to a normal distribution with $\mu=1000/6$ and $\sigma^2=5000/36.$
Thus $P(x\ge170)=1-P(x<170)=1-P(z<.2835)=.3936$
- And $P(162\le x\le 171)=P(-.3960<z<.3680)=P(z<.3680)-P(z>-.3960)=.6368-[1-P(z<-.3960)]=.6368-[1-.3520]<0$ which is not correct, must be between $0$ and $1$.
I have not checked everything, but there is an error in this line $$ P(-.3960<z<.3680)=P(z<.3680)-P(z>-.3960) $$ The final term should be $$ P(z\lt-.3960) $$ i.e. $$ P(-.3960\lt z \lt .3680)=P(z \lt .3680)-P(z\lt-.3960) $$