Let’s summarise everything we have done so far:
| Match | Prize | Probability % | Expected Value ($) |
| Jackpot | $20,000,000 | 0.00000034 | 0.068 |
| Five whites | $1,000,000 | 0.0000085 | 0.085 |
| Four whites + red | $50,000 | 0.0001095 | 0.055 |
| Four whites | $100 | 0.00274 | 0.0027 |
| Three whites + red | $100 | 0.006899 | 0.007 |
| Three whites | $7 | 0.1724838 | 0.012 |
| Two whites + red | $7 | 0.1425866 | 0.01 |
| One white + red | $4 | 1.087223 | 0.04 |
| Red | $4 | 2.609335 | 0.1 |
| Nothing | -$2 | 95.98 | -1.9196 |
| Total | 4.02 | -1.54 |
So, there is a 95.98% chance that you win nothing, and the expected value of the affair is -$1.54. To remind you, the expected value is the money I can make in the long run.
Not to forget, if the number of tickets sold increases, there could also be multiple people winning it, splitting the prize. Here is the probability of winning the jackpot estimated using the binomial distribution.
Here is the R code of the above plot:
xx <- seq(0, 500, 50)
yy <- dbinom(x = 0, size = xx, prob = 1/292.201338)
yy1 <- dbinom(x = 1, size = xx, prob = 1/292.201338)
yy2 <- dbinom(x = 2, size = xx, prob = 1/292.201338)
yy3 <- dbinom(x = 3, size = xx, prob = 1/292.201338)
yy4 <- 1 - pbinom(3, size = xx, prob = 1/292.201338)
par(bg = "antiquewhite1")
plot(xx,yy, ylim = c(0,1), type ="l", col = "red", xlab = "Number of tickets sold (million)", ylab="Probability")
polygon(x = c(0, xx, 500), y = c(0, yy, 0), col = "#006666")
text(x = 400, y = 0.1, "0 winner", col = "red")
lines(xx,yy+yy1, col ="blue" )
polygon(c(xx, rev(xx)), c(yy+yy1, rev(yy)),col = "#009999")
text(x = 400, y = 0.45, "1 winner", col = "red")
lines(xx,yy+yy1+yy2, col = "green" )
polygon(c(xx, rev(xx)), c(yy+yy1+yy2, rev(yy+yy1)), col = "#00cccc")
text(x = 400, y = 0.75, "2 winners", col = "red")
lines(xx,yy+yy1+yy2+yy3, col = "brown" )
polygon(c(xx, rev(xx)), c(yy+yy1+yy2+yy3, rev(yy+yy1+yy2)), col = "#00ffff")
text(x = 400, y = 0.92, "3 winners", col = "red")
lines(xx,yy+yy1+yy2+yy3+yy4, col = "black" )
polygon(c(xx, rev(xx)), c(yy+yy1+yy2+yy3+yy4, rev(yy+yy1+yy2+yy3)), col = "#99ffff")
text(x = 400, y = 0.98, ">3 winners", col = "red")
