Question

Give me answer in text with clear writing.

I roll a 6-sided die exactly once. Are the events “I roll an even number” and “I roll a number greater than 2” independent? Why or why not?

Answer 1

Suppose two events are A and B. These tow events are said to be independent if ,

that means prob(occurrence of the intersection of that two events)= prob(occurrence of A) * prob(occurrence of B)

Here the two events are,

roll an even number i.e rolling of (2,4,6) and roll a number greater than 2 i.e. rolling of (3,4,5,6)

Let us define that two events as A and B respectively

So, P(A)=P(occurring of 2,4 and 6)=P(2)+P(4)+P(6)=1/6 +1/6 +1/6 =3/6 = 1/2

[* If we throw a die once,then probability of comming every unit is 1/6]

Now, P(B)=P(occurring of 3,4,5,6)=P(3)+P(4)+P(5)+P(6) =1/6 +1/6+ 1/6 +1/6 =4/6 = 2/3

Now, means both the event will occur simulteneously ,So we need a case which is even and also greater than 2. So, will be occurrence of 4 and 6.

So, = P(occurrence of 4 and 6)=P(4)+P(5)=1/6 +1/6 =2/6 = 1/3

And P(A)*P(B) = 1/2 * 2/3 =1/3 =

As, we see that , Hence "I roll a even number" and "I roll a number greater than 2" are independent.

Hope this will help you,still if there is any doubt feel free to ask in comment section.