Help on discrete variables

Question

I need help with interpreting sentences in discrete random variables and how to convert it to tables,the question goes "A fair die is thrown once.A random variable represents the score on the uppermost face of the die.If the score is $two$ or more,then the random variable $X$ is the score.If the score is one,the die is to thrown once again and the random variable $X$ is the sum of the scores of the two throws.Construct the probability distribution table for $X$. "

Answer 1

The result of the first throw can be $1,2,3,4,5,6$ with equal probabilities. If the result of the first throw is $1$, then we throw the dice again and can obtain $1,2,3,4,5,6$ with equal probabilities. The random variable can be equal to $2,3,4,5,6$ with the probabilities $1/6$ after the first throw or it can be equal to $2,3,4,5,6,7$ with probability $(1/6)^2$ after the second throw. There are two ways to obtain $2,3,4,5,6$ and the probability is equal to $1/6+(1/6)^2=7/36$. Hence, the probability distribution table looks like this $$ \left.\begin{array}{c|c|c|c|c|c|c|c}\text{Probability} & 7/36 & 7/36 & 7/36 & 7/36 & 7/36 & 1/36 \\\hline \text{Value} & 2 & 3 & 4 & 5 & 6 & 7 \end{array}\right.. $$

Answer 2

Comment: Simulation of a million plays of this game in R verifies the answer by @Cm7F7Bb.

 B = 10^6
 d1 = sample(1:6, B, rep=T)
 d2 = sample(1:6, B, rep=T)
 s = numeric(B)              # vector of all 0's to start
 s[d1 > 1] = d1[d1 > 1]      # score if first die shows > 1
 s[d1 == 1] = (d1+d2)[d1==1] # score if first die shows 1

 table(s)/B
 ## s
 ##        2        3        4        5        6        7 
 ## 0.194416 0.194486 0.194510 0.194115 0.194735 0.027738 

 36*table(s)/B
 ## s
 ##        2        3        4        5        6        7 
 ## 6.998976 7.001496 7.002360 6.988140 7.010460 0.998568 

A very elementary way is just to list all 36 possible possibilities for two dice, putting the score in each of the 36 spaces in a table. Then count how many times each score appears.

 2nd \ 1st   1   2   3   4   5   6
 ----------------------------------
  1          2   2   3   4   5   6
  2          3   2   3   4   5   6
  3          4   2   3   4   5   6
  4          5   2   3   4   5   6
  5          6   2   3   4   5   6
  6          7   2   3   4   5   6
 ----------------------------------