Question

A die has six faces numbered from 1 to 6. It rolled and the number on the top face is noted. When this is treated as a random trial.
What are the possible outcomes?
Are they equally likely? Why?
Find the probability of a composite number turning up on the top face.

Answer 1

Hint: In probability theory, an outcome is a possible result of an experiment or trial equally likely. Outcomes are of those events in a sample space who have the same chances of accruing.
Probability of an outcome $=\dfrac{\text{No}\text{. of favorable outcomes}}{\text{No}\text{. of total possible outcomes}}$.

Complete step-by-step answer:
An experiment is said to be random if it has more than one possible outcome i.e. more than one possible value that we can get as a result.
Given a random trial is tossing a die. If we toss a die, the outcome will be getting a number from 1 to 6. So, the possible outcomes are $=\left\{ 1,2,3,4,5,6 \right\}$ ,
Where 1 = probability of getting 1 on top face,
2 = probability of getting 2 on top face,
.
.
.
.
6 = probability of getting 6 on the top face.
We know, Probability of an outcome $=\dfrac{\text{No}\text{. of favorable outcomes}}{\text{No}\text{. of total possible outcomes}}$ ,
${{P}_{1}}$ = Probability of getting ‘1’ on top face $=\dfrac{1}{6}$ ,
${{P}_{2}}$ = Probability of getting ‘2’ on top face $=\dfrac{1}{6}$ ,
Similarly, ${{P}_{3}}={{P}_{4}}={{P}_{5}}={{P}_{6}}=\dfrac{1}{6}$.
Equally likely means that each outcome of an experiment occurs with the same probability.
 Here, ${{P}_{1}}={{P}_{2}}={{P}_{3}}={{P}_{4}}={{P}_{5}}={{P}_{6}}=\dfrac{1}{6}$.
So, all the outcomes are equally likely. Now, we have to find the probability of the composite number turning up on the top face.
Total outcomes $=\left\{ 1,2,3,4,5,6 \right\}$ .
We know a composite number is defined as the number which has at least three factors. Out of the possible outcomes, composite numbers are 4 and 6.
So, total no of favorable outcomes for getting a composite number on top face = 2 and total outcomes =6, and we know, Probability of an outcome $=\dfrac{\text{No}\text{. of favorable outcomes}}{\text{No}\text{. of total possible outcomes}}$ .
So, probability of a composite number turning up on top face $=\dfrac{2}{6}$ .
$=\dfrac{1}{3}$ .
Note: Please note that for finding the probability of getting a composite number, we have added probability of getting a ‘4’ and the probability of getting ‘6’. We have done this using “Addition theorem of probability” which states that “if A and B are any two events” then the probability of happening of at least one of the events is defined as “$P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right)$ “.