I have this problem:
On each bet, a gambler loses 1 with probability 0.7, loses 2 with probability 0.2, or wins 10 with probability 0.1. Approximate the probability that the gambler will be losing after his first 100 bets.
My attempt:
Let $X_i$ be a random variable that denote the gambler’s winnings on the i-th bet.
Now,
$E(X) = -1(0.7) - 2(0.2)+10(0.1)=-0.1$, $E(X^2) = (-1)^2(0.7) - 2^2(0.2)+10^2(0.1)=11.5, Var(X) = 11.5-(-0.1)^2 = 11.49$
Now, we are interested in the first 100 bets, thus,
$E(\sum_{i=1}^{100} X_i) = \sum_{i=1}^{100} E(X_i) = 100(-0.1) = -10$ and $Var(\sum_{i=1}^{100} X_i) = \sum_{i=1}^{100} Var(X_i) = 100(11.49) = 1,149$
Therefore,
$P(X<0) = P(\frac{X+10}{\sqrt{1,149}}<\frac{10}{\sqrt{1,149}}) = P(Z<\frac{10}{\sqrt{1,149}}) = \Phi(0.295) = \Phi(0.295)=0.6141$
Am I right? I think that there is a problem saying that $P(Z<\frac{-10}{\sqrt{1,149}}) = \Phi(-0.295)$ because $P(Z\leq x) = \Phi(x)$