Question 4.35

4.35 Roulette winnings.

In the game of roulette, a wheel is spun and you place bets on where it will stop. One popular bet is that it will stop on a red slot; such a bet has an 18/38 chance of winning. I it stops on red, you double the money you bet. If not, you lose the money you bet. Suppose you play 3 times, each time with a $1 bet. Let Y represent the total amount won or lost. Write a probability model for Y.

I chose to use a Binomial Distribution to develop my model since the binomial distribution is used to describe the number of successes in a fixed number of trials.

For three trials, there are four outcomes:

1. you lose 3 times (-3$)
   
2. you lose 2 times and win 1 time (-1$)
   
3. you lose 1 time and win 2 times (+1$)
   
4. you lose 0 times and win 3 times (+3$)

success <- 0:3
odds <- dbinom(success, size = 3, prob=.474)
value <- c(-3,-1,1,3)
winnings <- c("-3$", "-1$", "1$","3$")
pocket_amt <- c("-3$", "2$", "4$","6$")
roulette <- data.frame(pocket_amt,winnings,value,odds)

kable(roulette, "html") %>% kable_styling("striped") %>% scroll_box(width = "100%", height = "100%")

pocket_amt	winnings	value	odds
-3$	-3$	-3	0.1455316
2$	-1$	-1	0.3934333
4$	1$	1	0.3545387
6$	3$	3	0.1064964

Let’s see what this looks like as plotted

barplot(roulette$odds, main = "Binomoal Distribution for Betting on Red for Three Roulette Spins",
        names.arg=c(winnings))

roulette[,3:4] <- sapply(roulette[,3:4], as.numeric)
win <-  roulette %>% filter(value >0) %>% summarise(odds = sum(odds))
lose <-  roulette %>% filter(value < 0) %>% summarise(odds = sum(odds))
odds <- rbind(win,lose)
outcome <-rbind("win", "lose")
win_lose <- cbind(outcome, odds)

kable(win_lose, "html") %>% kable_styling("striped") %>% scroll_box(width = "100%", height = "100%")

outcome	odds
win	0.4610352
lose	0.5389648

roulette[2] <- sapply(roulette[2], as.numeric)
win_lose <- win_lose %>% mutate(total = (odds * 100))

bp<- ggplot((win_lose), aes(x=NA, y=total, fill=outcome))+
geom_bar(width = 1, stat = "identity")
pie <- bp + coord_polar("y", start=0)
pie + scale_fill_grey() + theme_minimal()