Question

Roll a die and consider the following two events={2, 3, 6}, event ={1, 5, 6}. Are the events E and F independent?

Choose the correct answer below.

- No, because the probability of F occurring is higher if it is known that E has occurred.

- No, because there is at least one roll which leads to both E and F occurring.

- Yes, because the probability of F occurring is higher if it is known that E has occurred.

- Yes, because each roll of the die is an independent event.

Answer 1

Answer: The correct option is

(B) No, because there is at least one roll which leads to both E and F occurring.

Step-by-step explanation: We are given to roll a die and consider the following two events :

A = {2, 3, 6} and B = {1, 5, 6}.

We are to check whether the events A and B are independent.

We have

A ∩ B = {2, 3, 6} ∩ {1, 5, 6} = {6}.

So, n(A) = 3, n(B) = 3 and n(A ∩ B) = 1.

Since there are 6 elements in the sample space, we get

S = {1, 2, 3, 4, 5, 6} ⇒ n(S) = 6.

So, the probabilities of the events A, B and A ∩ B are calculated as follows :

`P(A)=(n(A))/(n(S))=(3)/(6)=(1)/(2),\\\\\\P(B)=(n(B))/(n(S))=(3)/(6)=(1)/(2),\\\\\\P(A\cap B)=(n(A\cap B))/(n(S))=(1)/(6).`

Therefore,

`P(A\cap B)=(1)/(6)\\eq P(A)* P(B)=(1)/(2)*(1)/(2)=(1)/(4)`

Thus, the events A and B are NOT independent because at least one roll which leads to both E and F occurring and that is 6, common to both A and B.

Option (B) is CORRECT.

Answer 2

Option 1 - No, the probability of F occurring is higher if it is known that E has occurred.

Explanation:

Given : Roll a die and consider the following two events={2, 3, 6}, event ={1, 5, 6}.

To find : Are the events E and F independent?

Solution :

E={2, 3, 6}, F={1, 5, 6}

They are not independent event as 6 is common in both the events.

Now, verifying by applying independent property,

Probability of event E is
`P(E)=(3)/(6)=(1)/(2)`

Probability of event F is
`P(F)=(3)/(6)=(1)/(2)`

Probability of E and F is
`P(E\cap F)=(1)/(6)`

Probability of F occuring when it is known that e has occured.

i.e.
`P(F/E)=(P(E\cap F))/(P(E))`

`P(F/E)=((1)/(6))/((1)/(2))`

`P(F/E)=(1)/(3)`

`P(F)>P(F/E)`

i.e. The probability of F occurring is higher if it is known that E has occurred.

Therefore, The correct option is 1.

Events E and F are not independent, because the probability of F occurring is higher if it is known that E has occurred.