Obtain the Probability Distribution of the Random Variable $( 1,1 ) ( 1,2 ) ( 1,3 ) ( 1,4 ) ( 1,5 ) ( 1,6 )$

question 124

Multiple Choice

Obtain the probability distribution of the random variable.
-When two balanced dice are rolled, 36 equally likely outcomes are possible as shown below. $( 1,1 ) ( 1,2 ) ( 1,3 ) ( 1,4 ) ( 1,5 ) ( 1,6 )$
$( 2,1 ) ( 2,2 ) ( 2,3 ) ( 2,4 ) ( 2,5 ) ( 2,6 )$
$( 3,1 ) ( 3,2 ) ( 3,3 ) ( 3,4 ) ( 3,5 ) ( 3,6 )$
$( 4,1 ) ( 4,2 ) ( 4,3 ) ( 4,4 ) ( 4,5 ) ( 4,6 )$
$( 5,1 ) ( 5,2 ) ( 5,3 ) ( 5,4 ) ( 5,5 ) ( 5,6 )$
$( 6,1 ) ( 6,2 ) ( 6,3 ) ( 6,4 ) ( 6,5 ) ( 6,6 )$

Let $X$ denote the product of the two numbers. Find the probability distribution of $X$ . Leave your probabilities in fraction form.

• A)
• \(\begin{array} { r | r r | r }
• \mathrm { x } & \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & \mathrm { x } & \mathrm { P } ( \mathrm { X } = \mathrm { x } ) \\
• \hline 2 & 1 / 18 & 10 & 1 / 12 \\
• 3 & 1 / 18 & 12 & 1 / 9 \\
• 4 & 1 / 12 & 15 & 1 / 12 \\
• 5 & 1 / 18 & 18 & 1 / 12 \\
• 6 & 1 / 9 & 20 & 1 / 12 \\
• 8 & 1 / 18 & 24 & 1 / 12 \\
• & & 30 & 1 / 18
• \end{array}\)
• B)
• \(\begin{array}{r|r}
• \mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \\
• \hline 1 & 1 / 18 \\
• 2 & 1 / 18 \\
• 3 & 1 / 18 \\
• 4 & 1 / 18 \\
• 5 & 1 / 18 \\
• 6 & 1 / 18 \\
• 8 & 1 / 18 \\
• 9 & 1 / 18 \\
• 10 & 1 / 18
• \end{array}\)
• \(\begin{array}{r|r}
• \mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \\
• \hline 12 & 1 / 18 \\
• 15 & 1 / 18 \\
• 16 & 1 / 18 \\
• 18 & 1 / 18 \\
• 20 & 1 / 18 \\
• 24 & 1 / 18 \\
• 25 & 1 / 18 \\
• 30 & 1 / 18 \\
• 36 & 1 / 18
• \end{array}\)
• C)
• \(\begin{array}{r|rr|r}
• \mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) & \mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \\
• \hline 1 & 1 / 36 & 12 & 1 / 9 \\
• 2 & 1 / 18 & 15 & 1 / 18 \\
• 3 & 1 / 18 & 16 & 1 / 36 \\
• 4 & 1 / 12 & 18 & 1 / 18 \\
• 5 & 1 / 18 & 20 & 1 / 18 \\
• 6 & 1 / 9 & 24 & 1 / 18 \\
• 8 & 1 / 18 & 25 & 1 / 36 \\
• 9 & 1 / 36 & 30 & 1 / 18 \\
• 10 & 1 / 18 & 36 & 1 / 36
• \end{array}\)
• D)
• \(\begin{array}{r|r}
• \mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \\
• \hline 2 & 1 / 36 \\
• 3 & 1 / 18 \\
• 4 & 1 / 12 \\
• 5 & 1 / 9 \\
• 6 & 5 / 36
• \end{array}\)
• \(\begin{array}{r|r}
• \mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \\
• \hline 7 & 1 / 6 \\
• 8 & 5 / 36 \\
• 9 & 1 / 9 \\
• 10 & 1 / 12 \\
• 11 & 1 / 18 \\
• 12 & 1 / 36
• \end{array}\)

---

Definitions:

Unpleasant Consequence

A negative result or effect that follows an action or decision, often serving as a deterrent for certain behaviors.

Punishment

The imposition of a penalty as a response to an offense or undesirable behavior, aiming to correct or deter such actions.

Operant Conditioning

A learning process in which the strength of a behavior is modified by reinforcement or punishment.

Pleasant Consequence

A positive outcome or reward that follows a desired behavior, often used in reinforcing such behavior.