Question

A fair die with numbers 1 to 6 on their faces is thrown. Let $X$ denote the number of factors of the number, on the uppermost face, then the probability distribution of $X$ is

• A. X=x: 1, 2, 3, 4; P(X=x): 1/6, 1/2, 1/6, 1/6
• B. X=x: 1, 2, 3, 4; P(X=x): 1/6, 1/6, 1/6, 1/2
• C. X=x: 1, 2, 3, 4; P(X=x): 1/6, 1/6, 1/6, 1/6
• D. X=x: 1, 2, 3, 4; P(X=x): 1/6, 1/6, 1/2, 1/6

Hint:

When constructing a probability distribution for a random variable defined on outcomes of a die roll, first list all possible outcomes of the die. Then, for each outcome, determine the value of the random variable. Finally, group identical values of the random variable and sum their probabilities.

Answer 1

The Correct Option is A

Step 1: Identify the sample space and calculate factors for each outcome.
A fair die is thrown, so the sample space is: \[ S = \{1, 2, 3, 4, 5, 6\} \] Each outcome has probability $\frac{1}{6}$. Let $X$ be the number of factors of the number on the uppermost face.

• For 1: Factors $\{1\}$, so $X(1) = 1$
• For 2: Factors $\{1, 2\}$, so $X(2) = 2$
• For 3: Factors $\{1, 3\}$, so $X(3) = 2$
• For 4: Factors $\{1, 2, 4\}$, so $X(4) = 3$
• For 5: Factors $\{1, 5\}$, so $X(5) = 2$
• For 6: Factors $\{1, 2, 3, 6\}$, so $X(6) = 4$

Step 2: Determine the possible values of the random variable $X$.
From Step 1, possible values of $X$ are: \[ 1, 2, 3, 4 \]
Step 3: Calculate probabilities.
For $X = 1$: \[ P(X=1) = \frac{1}{6} \] For $X = 2$ (comes from 2, 3, 5): \[ P(X=2) = \frac{3}{6} = \frac{1}{2} \] For $X = 3$ (comes from 4): \[ P(X=3) = \frac{1}{6} \] For $X = 4$ (comes from 6): \[ P(X=4) = \frac{1}{6} \] 
Step 4: Probability Distribution Table
 

$X$	1	2	3	4
$P(X)$	$\frac{1}{6}$	$\frac{1}{2}$	$\frac{1}{6}$	$\frac{1}{6}$