Question

You roll a six sided die. Find the probability of each of the following scenarios.

A) Rolling a 5 or a number greater than 3

B) Rolling a number less than 4 or an even number

C) Rolling a 6 or an odd number.

A) P (5 or number > 3)=______

B) P (1 or 2 or 3 or 4 or 6)=________

C) P (6 or 1 or 3 or 5)=________

Type an intger or a simplified fraction.......

Answer 1

Concepts and reason

The concept of probability and sample space is used in this problem.

Probability of any event is the likelihood of that event occurring in a random experiment.

Sample space is the collection of all the elements in a sample.

The probability of an event lies between 0 and 1. It should be positive and can take the form of fractions, decimals between 0 and 1. It can take the form of percentages between 1to100.

Fundamentals

The probability of an event A is calculated as:

$P\left( A \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}$

The addition rule of probability can be defined as:

$P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)$

(A)

The sample space of rolling a six sided die once is,

$S = \left\{ {1,2,3,4,5,6} \right\}$

All the events of rolling a six sided die is equally likely. That is, the probability of any number appearing on the face is same for each number.

The probability of rolling a 5 can be calculated as:

$\begin{array}{c}\\P\left( {{\rm{Rolling a 5 on a six sided die}}} \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}\\\\ = \frac{1}{6}\\\end{array}$ .

The sample space of rolling a number greater than 3 is,

$S = \left\{ {4,5,6} \right\}$

Therefore, the probability of rolling a number greater than 3 is,

$\begin{array}{c}\\P\left( {{\rm{Rolling a number greater than 3}}} \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}\\\\ = \frac{3}{6}\\\end{array}$

The sample space of rolling a 5 and a number greater than 3 is,

${S_1} = \left\{ 5 \right\}$

The probability of rolling a 5 and a number greater than 3 is,

$\begin{array}{c}\\P\left( {{\rm{Rolling a 5 and a number greater than 3}}} \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}\\\\ = \frac{1}{6}\\\end{array}$

The probability of rolling a 5 or a number greater than 3 is,

$\begin{array}{c}\\P\left( {5 \cup {\rm{a number greater than 3}}} \right) = \left[ \begin{array}{c}\\P\left( {\rm{5}} \right) + P\left( {{\rm{a number greater than 3}}} \right)\\\\ - P\left( {5 \cap {\rm{a number greater than 3}}} \right)\\\end{array} \right]\\\\ = \frac{1}{6} + \frac{3}{6} - \frac{1}{6}\\\\ = \frac{3}{6}\\\\ = \frac{1}{2}\\\end{array}$

(B)

The sample space of rolling a number less than 4 is,

$S = \left\{ {1,2,3} \right\}$

The probability of rolling a number less than 4 is,

$\begin{array}{c}\\P\left( {{\rm{Rolling a number less than 4 }}} \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}\\\\ = \frac{3}{6}\\\end{array}$

The sample space of rolling an even number is,

${S_1} = \left\{ {2,4,6} \right\}$

The probability of rolling an even number is,

$\begin{array}{c}\\P\left( {{\rm{Rolling an even number}}} \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}\\\\ = \frac{3}{6}\\\end{array}$

The sample space of rolling a number less than 4 and an even number is,

${S_2} = \left\{ 2 \right\}$

The probability of rolling a number less than 4 and an even number is,

$\begin{array}{c}\\P\left( {{\rm{Rolling a number less than 4 and an even number}}} \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}\\\\ = \frac{1}{6}\\\end{array}$

The probability of rolling a number less than 4 or an even number is,

$\begin{array}{c}\\P\left( {4 \cup {\rm{an even number}}} \right) = P\left( {\rm{4}} \right) + P\left( {{\rm{an even number}}} \right) - P\left( {4 \cap {\rm{an even number}}} \right)\\\\ = \frac{3}{6} + \frac{3}{6} - \frac{1}{6}\\\\ = \frac{5}{6}\\\end{array}$

(C)

The probability of rolling a 6 is,

$\begin{array}{c}\\P\left( {{\rm{Rolling a 6 on a six sided die}}} \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}\\\\ = \frac{1}{6}\\\end{array}$ .

The sample space of rolling an odd number is,

$S = \left\{ {1,3,5} \right\}$

The probability of rolling an odd number is,

$\begin{array}{c}\\P\left( {{\rm{Rolling an odd number}}} \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}\\\\ = \frac{3}{6}\\\end{array}$

The sample space of rolling a 6 and an odd number is,

${S_2} = \phi$

The probability of rolling a 6 and an odd number is,

$\begin{array}{c}\\P\left( {{\rm{Rolling a 6 and an odd number}}} \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}\\\\ = \frac{0}{6}\\\\ = 0\\\end{array}$

The probability of rolling a 6 or an odd number is :

$\begin{array}{c}\\P\left( {6 \cup {\rm{an odd number}}} \right) = P\left( {\rm{6}} \right) + P\left( {{\rm{an odd number}}} \right) - P\left( {6 \cap {\rm{an odd number}}} \right)\\\\ = \frac{1}{6} + \frac{3}{6} - 0\\\\ = \frac{4}{6}\\\\ = \frac{2}{3}\\\end{array}$

Ans: Part A

The probability of rolling a 5 or a number greater than 3 is $\frac{1}{2}$ .