A man goes to the casino and decides to simulate large sample in real life. The wheel has 38 pockets. 36 of which are numbered 1 to 36 and two others are numbered 0 and 00. The man places all his bets upon number 20 (doesn't matter, just any number). If he hits his number, the pay out is 35 to 1. For 365 nights, the man spins the wheel 200 times with $5 bet on number 20 for all of those spins. Not surprisingly, for each $1000 night, the man had mean loss of $55 each of those 365 nights. (this next part confused me but this is how book says it) The surprise, according to man, was the extreme variability of nightly winnings. 7 out of the 365 evenings, he lost the $1000 stake and only once did he win maximum of $1160. On 141 nights, the loss exceeded $250.
1) To measure the results of the man's experiment, first find the probability distribution of the gain x on a single $5 bet.
2) Find the expected value and variance of the gain x from part 1.
3) Find the expected value and variance for the evening's gain, the sum of the gains or losses for the 200 bets of $5 each.
4) Use the results of part 2 to evaluate the probability of 7 out of 365 evenings resulting in a loss of the total $1000 stake.
5) Use the results from part 3 to evaluate the probability that the largest evening's winnings were as great as $1160