Question

An experiment has $10$equally liked outcomes. Let A and B be two non-empty events of the experiment. If A consists of $4$outcomes, the number of outcomes B must have so that A and B are independent, is
A) $2$, $4$ or $8$
B) $3$, $6$ or $9$
C) $4$ or $8$
D) $5$ or $10$

Answer 1

Hint: First, we shall learn about the independent events. An independent event is an event that does not depend on the probability of another event. That is, two events A and B are said to be independent events if the probability of B does not affect the probability of A.
It can be understood by an example. Let us consider a probability of tossing a head and a probability of occurring Friday. Here the probability of tossing a head does not change the probability of tossing a head. Also, the probability of tossing a head does not change the probability of tossing a head. Hence, both the events have no connection between them. These events are known as independent events.

Formula used:
The two events are said to be independent events if
$P\left( {A \cap B} \right) = P\left( A \right) \times P\left( B \right)$
Where, $P\left( A \right)$is the probability of event A,
$P\left( B \right)$is the probability of event B and
$P\left( {A \cap B} \right)$is the probability of event A and event B.

Complete step by step answer:
Let A and B be two non-empty events.
Let us consider $P\left( A \right)$be the probability of event A.
Given that the number of outcomes of event A is $4$.
So, the probability of A \[ = \dfrac{{number{\text{ }}of{\text{ }}outcomes{\text{ }}of{\text{ }}A}}{{Total{\text{ }}number{\text{ }}of{\text{ }}outcomes}}\]
Here, the total number of outcomes is $10$.
Hence, the probability of event A =$\dfrac{4}{{10}} = \dfrac{2}{5}$
That is, $P\left( A \right) = \dfrac{2}{5}$
Let P be the number of outcomes of B.
Let us consider $P\left( B \right)$be the probability of event B.
So, the probability of B \[ = \dfrac{{number{\text{ }}of{\text{ }}outcomes{\text{ }}of{\text{ }}B}}{{Total{\text{ }}number{\text{ }}of{\text{ }}outcomes}}\]
The two events are said to be independent events if
$P\left( {A \cap B} \right) = P\left( A \right) \times P\left( B \right)$
Now we shall substitute the probability of A and B in the above formula.
Then $P\left( {A \cap B} \right) = \dfrac{2}{5} \times \dfrac{P}{{10}}$
We can also write that $P\left( {A \cap B} \right) = \dfrac{{\dfrac{{2P}}{5}}}{{10}}$
Here, $\dfrac{{2P}}{5}$ is the number of outcomes of $P\left( {A \cap B} \right)$
We all know that P is an integer because P is assumed as the number of outcomes and it must be a positive integer.
Now, we shall substitute $1,2,....$ in P in the expression $\dfrac{{2P}}{5}$
When we substitute $1,2,3,4,6,7,8,9,..$ , we will get the fractional numbers.
So, $\dfrac{{2P}}{5}$ which is the number of outcomes of $P\left( {A \cap B} \right)$must be a positive integer.
Hence, P will be the multiples of $5$
When $P = 5$ , $\dfrac{{2P}}{5} = \;\dfrac{{2\left( 5 \right)}}{5} = 2$
When $P = 10$ , $\dfrac{{2P}}{5} = \;\dfrac{{2\left( {10} \right)}}{5} = 4$
Here we are asked to find the number of outcomes of B.

According to the given options, $5$ or $10$ will be the answer, and hence option D) is the answer.

Note:
An independent event is an event that does not depend on the probability of another event. That is, two events A and B are said to be independent events if the probability of B does not affect the probability of A.
Also, the number of outcomes is always a positive integer or a natural number and it cannot be a negative value.