(a) The earning of each game can be modeled as a Bernoulli random variable with mean 1/37 and variance 1368/1369. If N games are played, then the casino earnings are the sum of N iid Bernoulli random variable, which has a binomial distribution, or since N is large, by Central Limit Theorem, it is approximately normal.
So if X denotes the casino earnings, then P(X > 1000) = 0.5 means, by symmetry of the normal distribution, that X has a mean of 1000. Since the Bernoulli random variable has a mean of 1/37, the casino needs to play 37000 games.
(b) Assuming that N games are played and N is large enough such that CLT is applicable, then
where is standard normal