Probability

P(A) = n(A)/S(A)

P(A): Probability of an event A happening
n(A): Number of ways of an event A happening
n(S): Total number of ways

P is between 0 and 1.

P(A) = 0: Event A doesn't happen.
P(A) = 1: Event A always happens.

Set A as
getting a multiple of 3 from a fair die.
{3, 6}

The number of {3, 6} is 2.
So n(A) = 2.

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

Then P(A) = 2/6.

2/6 = 1/3

So 1/3 is the answer.

There are 3 blue marbles.
Then n(A) = 3.
(Set A as picking a blue marble.)

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

3 + 4 = 7

7 + 5 = 12

So n(S) = 12.

There are 3 blue marbles.
So n(A) = 3.

There are 12 marbles in the jar.
So n(S) = 12.

Then P(A) = 3/12.

3/12 = 1/4

So 1/4 is the answer.

P(X ≥ 4) means
the probability of
getting a number greater than 4.

So it means
the probability of getting 4 or 5.

4 and 5 area is the colored area.
The colored area is 2/5 of the whole spinner.

So the probability of getting 4 or 5 is
2/5.

So 2/5 is the answer.