Question

Select from the following pairs of events all the pairs that are independent.

Answer 1

Answer:

A.) Independent

B.) Independent

C.) Dependent

D.) Independent

E.) Dependent

F.) Independent

Step-by-step explanation:

To find - Select from the following pairs of events all the pairs that are independent.

A. Draw a 2 of clubs from a standard deck of 52 cards, keep it, then draw a 2 of diamonds.

B. Draw a 3 of spades from a standard deck of 52 cards, replace it, then draw a 5 of hearts.

C. Roll a number cube. Then roll again if the first roll is a 6.

D. Roll a 2 on a number cube and spin a 3 on a spinner.

E. Events A and B, where P(A) = 0.4, P(B) = 0.2, and P(A and B) = 0.8.

F. Events A and B, where P(A) = 0.1, P(B) = 0.5, and P(A and B) = 0.05.

Proof -

We know that,

Independent events are the events that does not depend on the previous events.

Also,

If A and B are two independent events, then

P(A and B) = P(A) * P(B)

Now,

A.)

Draw a 2 of clubs from a standard deck of 52 cards, keep it, then draw a 2 of diamonds.

Here,

Event 1 - Draw a 2 of clubs from a standard deck of 52 cards

Event 2 - keep it, then draw a 2 of diamonds.

These Events are independent

Because

Event 1 is about picking a card that is club

Event 2 is about picking a card that is diamond

And Diamond and spades are 2 different types of card in a pack of standard deck.

So, they are not related to each other.

So,

Both the Events are Independent.

B.)

Draw a 3 of spades from a standard deck of 52 cards, replace it, then draw a 5 of hearts.

Event 1 is about picking a card that is spade

Event 2 is about replacing the card to heart.

These Events are independent

Because

And spade and heart are 2 different types of card in a pack of standard deck.

So, they are not related to each other.

So,

Both the Events are Independent.

C.)

Roll a number cube. Then roll again if the first roll is a 6.

Event 1 is Roll a number cube.

Event 2 is Roll again if the first roll is 6.

These Events are Dependent

Because

Event 2 is dependent on Event 1.

If Event 1 given no 6 then Event 2 would not happen

If Event 1 given 6 then Event 2 would happen.

So, They are related to each other

So,

Both the Events are Dependent.

D.)

Roll a 2 on a number cube and spin a 3 on a spinner.

Event 1 is Roll a 2 on a number cube

Event 2 is spin a 3 on a spinner

These Events are independent

Because

Cube and spinner are two different objects.

So, they are not related to each other.

So,

Both the Events are Independent.

E.)

Events A and B, where P(A) = 0.4, P(B) = 0.2, and P(A and B) = 0.8.

These Events are Dependent

Because

P(A and B) = P(A)*P(B)

⇒0.8 = 0.4*0.2

⇒0.8 = 0.08

But they are not equal

So, they are not independent Events.

F.)

Events A and B, where P(A) = 0.1, P(B) = 0.5, and P(A and B) = 0.05.

These Events are independent

Because

P(A and B) = P(A)*P(B)

⇒0.05 = 0.1*0.5

⇒0.05 = 0.05

They are not equal

So, they are independent Events.