Chapter SUMS OF DISCRETE RANDOM VARIABLES

Exercise 1 Page 289

A die is rolled three times. Find the probability that the sum of the outcomes is:-

  1. greater than 9.

Solution

We know a dice has 6 sides. That means the number of possibilities per roll is 6. If we roll the dice 3 times. The total number of possible outcomes are \[ 6^3 = \] 216

n = 216
dice = seq(1,6)
gtrthnine = 0
for (i in 1:6) {      # First Roll
 for (j in 1:6) {     # Second Roll
   for (k in 1:6) {   # Third Roll
     if ((i+j+k) > 9) # Calculate if the sum is greater than 9
       gtrthnine = gtrthnine + 1
   }
  }

}

The number of total events where the sum of 3 rolls is greater than 9 is :- 135

Probability of the same is :- gtrthnine/n = 0.625

  1. an odd number.

Solution

Lets consider the probability if getting an odd for

n = 216
dice = seq(1,6)
odd = 0
for (i in 1:6) {      # First Roll
 for (j in 1:6) {     # Second Roll
   for (k in 1:6) {   # Third Roll
     if ((i+j+k)%%2 != 0) # Calculate if the sum is Odd
       odd = odd + 1
   }
  }

}

The number of total events where the sum of 3 rolls is greater than 9 is :- 108

Probability of the same is :- odd/n = 0.5