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.
