Probability: Independent Events

Set A as
getting a multiple of 3 from a fair die.
Then n(A) = 1.

A die has 6 sides.
So n(S) = 6.

Then P(A) = 1/6.

Probability

Set B as
getting a head from a coin.
Then n(B) = 1.

A coin has 2 sides.
So n(S) = 2.

Then P(B) = 1/2.

Getting a 3 from a die (A)
and
getting a head from a coin (B)
do not affect each other.

So A and B are independent events.

P(A) = 1/6
P(B) = 1/2

So
P(A and B) = [1/6]⋅[1/2].

[1/6]⋅[1/2] = 1/12

So 1/12 is the answer.

First, let's see the 1st pick.

There are 5, 4, and 3 marbles in the jar.
So n(S) = 5 + 4 + 3.

5 + 4 = 9

9 + 3 = 12

So n(S) = 12.

This means,
for the 1st pick,
there are 12 marbles.

There are 5 blue marbles.
So n(A) = 5.

n(S) = 12

So P(A) = 5/12.

Probability

Next, let's see the 2nd pick.

It says
a marble is randomly picked from the jar
and [replaced].

So the 1st pick (A)
doesn't affect the 2nd pick (B).

So A and B are independent events.

A and B are both
picking a blue marble
in the same condition.
(because of the replacement)

So P(A) and P(B) are the same.

So P(B) = P(A) = 5/12.

P(A) = 5/12
P(B) = 5/12

A and B are independent events.

So
P(A and B) = [5/12]⋅[5/12].

5⋅5 = 25
12⋅12 = 144

So 25/144 is the answer.

A and B are independent events.

P(A) = 4/7
P(B) is unknown.
P(A and B) = 1/7

Then
[1/4]⋅P(B) = 4/7.

Multiply 7 to both sides.

Then 4⋅P(B) = 1.

Divide both sides by 4.

Then P(B) = 1/4.

P(A) = 4/7
P(B) = 1/4
P(A and B) = 1/7

Then
P(A or B) = 4/7 + 1/4 - 1/7.

Probability: A or B

Change the denominators to 28.

[4/7]⋅[4/4] = 16/28
+[1/4]⋅[7/7] = +7/28
-[1/7]⋅[4/4] = -4/28

16/28 + 7/28 - 4/28 = 19/28

So 19/28 is the answer.