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Student Name : Sachid Deshmukh

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Chapter-1 Exercise-3

In the early 1600s, Galileo was asked to explain the fact that,although the number of triples of integers from 1 to 6 with sum 9 is the same as the number of such triples with sum 10, when three dice are rolled, a 9 seemed to comeup less often than a 10-supposedly in the experience of gamblers.

(a) Write a program to simulate the roll of three dice a large number of times and keep track of the proportion of times that the sum is 9 and the proportion of times it is 10.

(b) Can you conclude from your simulations that the gamblers were correct?

sumulateDice = function(n)
{
  set.seed(123)
  diceSum = list(0)
  for(i in 0:n)
  {
    dice1 = sample(1:6, size=1, replace=TRUE)
    dice2 = sample(1:6, size=1, replace=TRUE)
    dice3 = sample(1:6, size=1, replace=TRUE)
    sumDice = dice1 + dice2 + dice3
    diceSum[i] = sumDice
  }
  return (diceSum)
}

print("Three Dice Roll for 100 times")
## [1] "Three Dice Roll for 100 times"
diceSum = sumulateDice(100)

print(paste("Proportion of 9 = ",  length(diceSum[diceSum == 9])/length(diceSum)))
## [1] "Proportion of 9 =  0.14"
print(paste("Proportion of 10 = ", length(diceSum[diceSum == 10])/length(diceSum)))
## [1] "Proportion of 10 =  0.16"
print("Three Dice Roll for 1000 times")
## [1] "Three Dice Roll for 1000 times"
diceSum = sumulateDice(1000)

print(paste("Proportion of 9 = ",  length(diceSum[diceSum == 9])/length(diceSum)))
## [1] "Proportion of 9 =  0.131"
print(paste("Proportion of 10 = ", length(diceSum[diceSum == 10])/length(diceSum)))
## [1] "Proportion of 10 =  0.133"
print("Three Dice Roll for 10000 times")
## [1] "Three Dice Roll for 10000 times"
diceSum = sumulateDice(10000)

print(paste("Proportion of 9 = ",  length(diceSum[diceSum == 9])/length(diceSum)))
## [1] "Proportion of 9 =  0.1218"
print(paste("Proportion of 10 = ", length(diceSum[diceSum == 10])/length(diceSum)))
## [1] "Proportion of 10 =  0.1233"

Conclusion: From the three simulations above, we can conclude that proportions of 9 and 10 are almost the same. The difference is in-significant as the no of simulations increases