distribution, cumulative distribution function, density function, expected
Chapter 14. Roulette/2: normal
approximation (exercises)
For the roulette, see Freedman–Pisani–Purves: Statistics, pp. 281-286.
exercise set A (100 games)
1) Jean plays European roulette (with 37 slots) a hundred times, putting 1 dollar on Red each time. If he wins he gets his dollar back with winnings of 1 dollar. If he loses he gets nothing. (He wins with 18 of the 37 slots.) a) Find the chance of his 'being ahead' after the 100 games. (That is, the chance that the sum of what he gets exceeds the sum of what he pays.)
b) Find the chance of his net gain from the 100 games being over 30 percent of his bets. (That is, the chance that the sum of what he gets minus the sum of what he pays exceeds 30 dollars.)
2) Jean plays European roulette a hundred times, putting 1 dollar on First dozen each time. If he wins he gets his dollar back with winnings of 2 dollars. If he loses he gets nothing. (He wins with 12 of the 37 slots.)
a) Find the chance of his 'being ahead' after the 100 games. (That is, the chance that the sum of what he gets exceeds the sum of what he pays.)
b) Find the chance of his net gain from the 100 games being over 30 percent of his bets.
3) Jean plays European roulette a hundred times, putting 1 dollar on First row each time. If he wins he gets his dollar back with winnings of 11 dollars. If he loses he gets nothing. (He wins with 3 of the 37 slots.)
a) Find the chance of his 'being ahead' after the 100 games. (That is, the chance that the sum of what he gets exceeds the sum of what he pays.)
b) Find the chance of his net gain from the 100 games being over 30 percent of his bets.
exercise set B (1000 games)
1) Jean plays European roulette a thousand times, putting 1 dollar on Red each time.
a) Find the chance of his 'being ahead' after the 1,000 games.
b) Find the chance of his net gain from the 1,000 games being over 30 percent of his bets that is, over 300 dollars.
c) Find the chance of his net gain from the 1,000 games being over 30 dollars.
d) Find the chance of his net gain from the 1,000 games being over 95 dollars.
2) Jean plays European roulette a thousand times, putting 1 dollar on First dozen each time.
a) Find the chance of his 'being ahead' after the 1,000 games.
b) Find the chance of his net gain from the 1,000 games being over 30 percent of his bets that is, over 300 dollars.
c) Find the chance of his net gain from the 1,000 games being over 30 dollars.
d) Find the chance of his net gain from the 1,000 games being over 95 dollars.
3) Jean plays European roulette a thousand times, putting 1 dollar on First row each time.
a) Find the chance of his 'being ahead' after the 1,000 games.
c) Find the chance of his net gain from the 1,000 games being over 30 dollars.
d) Find the chance of his net gain from the 1,000 games being over 95 dollars.
exercise set C (10,000 games)
1) Jean plays European roulette ten thousand times, putting 1 dollar on Red each time.
a) Find the chance of his 'being ahead' after the 10,000 games.
b) Find the chance of his net gain from the 10,000 games being over 30 percent of his bets that is, over 3.000 dollars.
c) Find the chance of his net gain from the 10,000 games being over 30 dollars.
d) Find the chance of his net gain from the 10,000 games being over 300 dollars.
2) Jean plays European roulette ten thousand times, putting 1 dollar on First dozen each time.
a) Find the chance of his 'being ahead' after the 10,000 games.
b) Find the chance of his net gain from the 10,000 games being over 30 percent of his bets that is, over 3.000 dollars.
c) Find the chance of his net gain from the 10,000 games being over 30 dollars.
d) Find the chance of his net gain from the 10,000 games being over 300 dollars.
3) Jean plays European roulette ten thousand times, putting 1 dollar on First row each time.
a) Find the chance of his 'being ahead' after the 10,000 games.
b) Find the chance of his net gain from the 10,000 games being over 30 percent of his bets that is, over 3.000 dollars.
c) Find the chance of his net gain from the 10,000 games being over 30 dollars.
d) Find the chance of his net gain from the 10,000 games being over 300 dollars.
exercise set D
1) Find the expected value of the net gain from 100 games, putting 1 dollar (a) on Red (b) on First dozen (c) on First row
each time.
2) Find the standard deviation of the net gain from 100 games, putting 1 dollar (a) on Red (b) on First dozen (c) on First row
each time.
1’) Find the expected value of the net gain from 1,000 games, putting 1 dollar (a) on Red (b) on First dozen (c) on First row
each time.
2’) Find the standard deviation of the net gain from 1,000 games, putting 1 dollar
each time.
1”) Find the expected value of the net gain from 10,000 games, putting 1 dollar (a) on Red (b) on First dozen (c) on First row
each time.
2”) Find the standard deviation of the net gain from 10,000 games, putting 1 dollar (a) on Red (b) on First dozen (c) on First row
each time.
exercise set E
1) We want a net gain of at least 50%. (That is, spending 100 dollars, we want a net gain of at least 50 dollars.) Which gives you a better chance,
a) putting 1 dollar on Red a hundred times?
b) putting 1 dollar on First dozen a hundred times?
c) putting 1 dollar on First row a hundred times?
2) We want a net gain of at least 50%. (That is, spending 10,000 dollars, we want a net gain of at least 5.000 dollars.) Which gives you a better chance,
a) 100 games, putting 100 dollars on Red each time?
b) 1,000 games, putting 10 dollars on Red each time?
c) 10,000 games, putting 1 dollar on Red each time?
d) (1 game) putting all 10,000 dollars on Red at once?