How to win game with probability

Game with probability

There are many ways to win the game with probability.

 Let us want to flip a balanced coin multiple times using R

R-code

> outspc = c("Heads","Tails") #Outcome space > z = sample(outspc, 20, replace=TRUE)#Flip 20 times > z # 1ST TIME [1] "Tails" "Tails" "Heads" "Heads" "Heads" [6] "Heads" "Heads" "Tails" "Tails" "Heads" [11] "Heads" "Heads" "Heads" "Tails" "Tails" [16] "Heads" "Heads" "Tails" "Heads" "Heads" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #2ND TIMES [1] "Tails" "Tails" "Heads" "Tails" "Heads" [6] "Heads" "Tails" "Tails" "Tails" "Tails" [11] "Tails" "Tails" "Tails" "Tails" "Tails" [16] "Heads" "Tails" "Heads" "Heads" "Heads" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z # 3RD TIMES [1] "Tails" "Heads" "Heads" "Tails" "Heads" [6] "Heads" "Heads" "Heads" "Heads" "Tails" [11] "Heads" "Heads" "Heads" "Heads" "Tails" [16] "Tails" "Heads" "Heads" "Heads" "Tails" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z 4TH TIMES [1] "Tails" "Heads" "Tails" "Heads" "Heads" [6] "Heads" "Heads" "Heads" "Tails" "Tails" [11] "Tails" "Tails" "Tails" "Tails" "Heads" [16] "Heads" "Tails" "Tails" "Heads" "Tails" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #5TH TIMES [1] "Tails" "Heads" "Tails" "Tails" "Heads" [6] "Heads" "Tails" "Heads" "Tails" "Heads" [11] "Tails" "Heads" "Tails" "Heads" "Tails" [16] "Tails" "Tails" "Heads" "Tails" "Tails" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #6TH TIMES [1] "Heads" "Heads" "Heads" "Tails" "Heads" [6] "Heads" "Heads" "Heads" "Heads" "Tails" [11] "Heads" "Heads" "Tails" "Heads" "Heads" [16] "Heads" "Tails" "Heads" "Heads" "Tails" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #7TH TIMES [1] "Heads" "Heads" "Tails" "Heads" "Tails" [6] "Tails" "Heads" "Heads" "Heads" "Heads" [11] "Tails" "Tails" "Heads" "Tails" "Heads" [16] "Heads" "Heads" "Heads" "Tails" "Heads" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #8TH TIMES [1] "Heads" "Tails" "Heads" "Heads" "Heads" [6] "Tails" "Heads" "Heads" "Tails" "Tails" [11] "Heads" "Heads" "Heads" "Heads" "Heads" [16] "Tails" "Heads" "Heads" "Heads" "Heads" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #9TH TIMES [1] "Heads" "Tails" "Tails" "Heads" "Heads" [6] "Heads" "Tails" "Tails" "Tails" "Heads" [11] "Tails" "Heads" "Heads" "Heads" "Heads" [16] "Heads" "Heads" "Heads" "Tails" "Heads" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #10TH TIMES [1] "Heads" "Tails" "Tails" "Tails" "Heads" [6] "Tails" "Heads" "Tails" "Heads" "Heads" [11] "Tails" "Heads" "Heads" "Heads" "Heads" [16] "Tails" "Heads" "Tails" "Heads" "Heads" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #11TH TIMES [1] "Tails" "Heads" "Heads" "Tails" "Heads" [6] "Tails" "Tails" "Tails" "Heads" "Tails" [11] "Heads" "Heads" "Heads" "Tails" "Heads" [16] "Heads" "Heads" "Heads" "Heads" "Tails" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #12TH TIMES [1] "Heads" "Heads" "Tails" "Heads" "Heads" [6] "Tails" "Heads" "Tails" "Tails" "Heads" [11] "Tails" "Heads" "Heads" "Heads" "Tails" [16] "Tails" "Tails" "Heads" "Heads" "Heads" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #13TH TIMES [1] "Heads" "Tails" "Tails" "Heads" "Tails" [6] "Heads" "Tails" "Heads" "Heads" "Heads" [11] "Heads" "Heads" "Tails" "Tails" "Heads" [16] "Heads" "Tails" "Heads" "Heads" "Heads" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #14TH TIMES [1] "Heads" "Tails" "Tails" "Tails" "Tails" [6] "Tails" "Tails" "Heads" "Heads" "Heads" [11] "Tails" "Heads" "Heads" "Tails" "Tails" [16] "Tails" "Tails" "Heads" "Heads" "Tails" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #15TH TIMES [1] "Tails" "Heads" "Heads" "Heads" "Heads" [6] "Tails" "Heads" "Heads" "Heads" "Heads" [11] "Tails" "Heads" "Heads" "Heads" "Tails" [16] "Heads" "Heads" "Tails" "Tails" "Heads" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #16TH TIMES [1] "Heads" "Tails" "Tails" "Tails" "Tails" [6] "Tails" "Tails" "Tails" "Heads" "Heads" [11] "Tails" "Heads" "Heads" "Tails" "Tails" [16] "Heads" "Tails" "Heads" "Tails" "Tails" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #17TH TIMES [1] "Tails" "Heads" "Tails" "Heads" "Tails" [6] "Tails" "Tails" "Tails" "Heads" "Heads" [11] "Heads" "Heads" "Tails" "Heads" "Heads" [16] "Heads" "Heads" "Heads" "Tails" "Heads" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #18TH TIMES [1] "Heads" "Heads" "Tails" "Tails" "Tails" [6] "Heads" "Tails" "Heads" "Heads" "Heads" [11] "Heads" "Tails" "Heads" "Tails" "Heads" [16] "Heads" "Tails" "Tails" "Heads" "Tails" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #19TH TIMES [1] "Tails" "Tails" "Heads" "Tails" "Heads" [6] "Tails" "Tails" "Heads" "Heads" "Heads" [11] "Heads" "Heads" "Heads" "Tails" "Heads" [16] "Heads" "Tails" "Heads" "Tails" "Heads" > outspc = c("Heads","Tails") > z = sample(outspc, 20, replace=TRUE) > z #20TH TIMES [1] "Heads" "Heads" "Tails" "Heads" "Heads" [6] "Tails" "Tails" "Heads" "Tails" "Tails" [11] "Heads" "Tails" "Heads" "Tails" "Tails" [16] "Tails" "Tails" "Heads" "Tails" "Tails" But flipping the coin by hand is very time consuming so using R-code > n = 5000 > outc = sample(c("Head","Tail"), n, replace=T) > z = cumsum(outc=="Head")/seq(1,n) > plot(z, xlab="Flips", ylab="Frequency of Heads",type="l") > abline(h=0.5, col="grey") If We want to roll six sided die 15 times 1 St time > outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 6 2 6 6 1 2 6 6 2 6 5 3 1 4 2 2nd times > outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 3 2 4 4 4 1 2 6 1 5 3 5 2 1 4 3rd times > outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 2 3 5 4 6 4 6 1 1 5 3 3 2 6 4 4th times > outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 1 3 6 3 4 4 2 6 6 3 3 2 1 5 2 5th times > outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 6 6 4 3 4 5 4 3 3 4 1 2 2 4 6 6th times > outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 3 5 1 1 3 4 3 3 1 6 5 2 1 2 4 7th times > outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 5 6 3 3 5 4 5 5 3 1 2 1 2 5 5 8th times > outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 2 2 6 4 3 3 6 6 2 4 3 5 1 4 4 9th times > outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 3 5 5 6 6 1 3 6 3 5 1 3 4 1 5 10th times > outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 4 5 6 5 2 5 1 4 1 3 4 3 6 5 5 11th times > outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 5 6 3 4 1 1 6 3 5 3 3 5 3 3 4 12th times outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 6 5 4 1 1 2 2 5 1 5 2 4 1 4 4 13th times > outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 6 3 2 4 1 5 4 5 2 3 1 6 5 1 5 14th times > outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 6 6 6 1 2 1 6 6 3 2 1 2 4 6 1 15th times > outspc=seq(1,6) > y = sample(outspc, 15, replace=TRUE) > y [1] 6 4 3 5 4 4 4 4 2 3 3 5 3 4 1 Learn Probability PYTHON