1.) A roulette wheel has 38 slots, of which 18 are red, 18 are black, and 2 are
ID: 2960217 • Letter: 1
Question
1.) A roulette wheel has 38 slots, of which 18 are red, 18 are black, and 2 are green (see Examples 5.5 and 5.8). A bet on "red" pays even money. Thus, if you bet $1 on "red" and the ball ends up in a red slot, you get your dollar back, plus winnings of $1. But if the ball ends up in a black or green slot, you lose your $1 bet. We now decide to make this bet five times in a row. We wish to estimate the probability distribution of our total winnings (net gain) after these five bets.Design a box model for your net gain on a single $1 bet on “red”. Place the “values” in the table in ascending order (from smallest to largest), and place their corresponding counts in the space next to them. Use the “obvious” counts instead of reducing to the smallest possible counts.
count | value (in dollars)
-------------------------------
(A) | (B)
-------------------------------
(C) | (D)
What is the value of (A) in the above table?
What is the value of (B) in the above table?
What is the value of (C) in the above table?
What is the value of (D) in the above table?
2.) We plan to make this bet five times in a row and we are interested in our net gain over the five bets. What would we classify as one simulation of this experiment?
A) Draw n with replacement
B) Draw n without replacement
C) Draw once
D) Draw until all X values occur
E) Draw until specified X value occurs
F) Bootstrap
G) Randomize
H) Permute
3.) If you need to specify the value of n, what is it?
4.) What is the statistic of interest for this simulation experiment?
A) p-hat
B) mean
C) sum
D) standard deviation
E) median
F) interquartile range
G) min
H) max
I) D
J) sample size
K) the draw
L) chi-square
Explanation / Answer
How can you make any money on this? The only way to come away with any money is to win every time, and then we come away with a lot of money. How can you lose all of your money? Lots of ways. All you need to do is land on a green or a black once in the five times we bet. It doesn't matter which one of the five bets we lose. This makes things easier to calculate. 18/(18+18+2) is the probability that we win. To win five times in a row we raise this probability to the fifth power. (18/38)^5 = 0.0238 If we keep winning, we are paid 2, 4, 8, 16, 32, so if we always win, we get $32. Since we started with $1, our net gain is $31. This happens 2.38% of the time. All the rest of the time we start with $1 and end up with $0, so our net gain is -$1. The weighted average of these is $31(.0238) + -1*(.9762) = -0.236877039 so, on average, our net loss is twenty-four cents. count value .976 -1 .0238 31 A is 0.976 B is -1 C is 0.0238 D is 31 One simulation of this experiment could be classified as (B), draw n with replacement. This is how we found the 0.023847593, we took (18/38)^5, which is the same as draw five with replacement since the odds stayed the same for all five bets. (not totally sure on this answer, check your class notes to see if you professor had another term for this) If we need to specify the value of n it would be five. If we wish to estimate our net gain after the five bets, we want the average, or (B), the mean. (not totally sure on this answer, this is what I used to find the average 24 cent loss)
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