A tourist in Las Vegas was attracted by a certain gambling game in which the customer stakes 1 dollar on each play; a win then pays the customer 2 dollars plus the return of her stake, although a loss costs her only her stake. Las Vegas insiders, and alert students of probability theory, know that the probability of winning at this game is 1/4. When driven from the tables by hunger, the tourist had played this game 240 times.
Assuming that no near miracles happened, about how much poorer was
the tourist upon leaving the casino?
We need to find the expected value of a single game first:
When they win: 2 * 1/4
When they lose: -1 * 3/4
\(E(x) = 2*\frac{1}{4} +
-1*\frac{3}{4}\) => \(\frac{1}{2} -
\frac{3}{4}\) => \(-\frac{1}{4}\)
Then, we can multiply this by the 240 games
\(-\frac{1}{4} * 240 = -60\)
After 240 games, we expect the tourist is leaving $60 poorer.
What is the probability that she lost no money?
We can find this using 1 - P(x ≥ 0)
n <- 240
p <- 1/4
q <- 3/4
sd <- sqrt(n*p*q)
mean <- -60
cat("1 - P(x ≥ 0) = 1 -", pnorm(240, mean, sd), "= 0%\n")## 1 - P(x ≥ 0) = 1 - 1 = 0%print(rnorm(n, mean, sd))## [1] -53.91003 -63.49636 -72.20752 -49.71380 -57.29704 -75.57619 -58.57813
## [8] -62.90236 -49.57611 -68.25157 -54.33253 -44.81957 -50.44700 -62.14448
## [15] -51.01833 -52.85819 -46.15200 -64.80486 -58.83586 -52.15179 -60.18900
## [22] -58.40075 -63.43056 -55.71170 -66.91528 -65.35006 -62.33756 -58.64120
## [29] -58.12172 -59.78821 -42.95947 -67.61805 -62.15620 -57.09295 -63.09572
## [36] -57.94958 -54.36099 -62.49639 -61.73560 -66.57775 -59.83371 -61.33730
## [43] -64.51930 -62.08565 -68.60033 -50.92779 -71.90549 -62.06174 -70.34686
## [50] -59.10110 -57.62058 -62.61670 -67.86573 -58.92315 -61.28540 -63.91951
## [57] -68.76085 -62.68733 -62.06927 -61.55272 -68.58110 -71.35451 -57.42998
## [64] -63.14998 -62.92701 -58.14891 -68.09002 -62.87659 -61.69228 -58.90964
## [71] -66.83989 -58.98630 -52.05315 -58.09251 -61.94498 -62.98591 -54.31362
## [78] -67.43011 -67.52468 -61.39780 -61.40354 -66.43345 -60.53103 -61.99774
## [85] -59.46367 -54.71231 -49.77007 -61.82862 -50.19250 -68.11460 -54.33387
## [92] -63.36108 -60.41980 -66.22772 -56.53995 -67.71220 -53.94763 -61.21110
## [99] -61.83172 -63.04044 -53.60880 -63.53959 -61.24922 -63.94656 -58.06932
## [106] -71.46783 -62.34431 -49.53412 -58.45399 -57.48708 -61.24384 -54.35178
## [113] -64.98208 -70.54191 -58.00287 -57.29269 -65.75368 -56.67497 -54.75406
## [120] -60.96393 -65.71991 -51.09375 -46.80272 -64.98039 -59.84537 -49.18547
## [127] -61.01128 -61.25926 -67.87167 -60.73957 -61.07658 -53.90240 -64.72063
## [134] -61.29206 -53.88373 -58.48486 -58.17851 -59.40200 -59.02381 -63.27280
## [141] -55.93405 -53.75521 -51.88375 -57.35882 -55.80276 -59.08935 -53.06430
## [148] -71.22808 -44.60611 -70.03535 -56.94193 -61.52286 -54.42068 -62.51443
## [155] -61.30843 -72.36661 -53.32784 -52.04344 -50.35941 -68.60147 -65.51121
## [162] -53.96206 -58.26339 -62.94946 -50.59864 -68.26415 -51.09424 -70.50348
## [169] -65.71131 -70.20217 -67.49537 -67.93200 -61.27570 -65.08218 -59.93941
## [176] -58.61715 -74.39225 -51.01998 -70.97144 -69.69760 -66.17055 -56.67385
## [183] -53.42493 -59.47662 -64.22918 -55.17705 -55.99352 -55.64836 -64.14357
## [190] -51.33868 -72.77801 -57.41282 -64.08943 -63.72507 -69.89153 -57.83579
## [197] -75.07433 -67.89152 -56.10477 -62.32247 -78.07366 -63.39962 -65.17826
## [204] -59.22327 -57.68039 -61.43155 -47.32224 -61.67792 -42.28670 -50.95192
## [211] -74.90174 -58.40320 -52.75976 -64.56626 -63.92220 -59.69124 -68.76330
## [218] -64.03097 -52.29384 -63.50071 -57.43421 -56.04518 -57.88910 -63.03879
## [225] -65.70670 -77.37117 -64.94442 -50.30351 -64.43437 -54.01780 -75.46253
## [232] -60.23978 -56.18214 -51.92403 -44.90083 -54.74229 -50.40879 -44.14639
## [239] -48.03855 -48.67504We can use the data for the rnorm command above to check as it produces a list of 240 different outcomes of the amount of money lose or gain. Since there is no 0 or positive gain, there is no way of getting P(x ≤ 0).