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# Probability (Math)

See how to find the probability of an event.
15 examples and their solutions.

## Probability

### Formula

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

P(A): 0 ~ 1
P(A) = 0: A does not happen.
P(A) = 1: A always happens.

### Example

A fair die is tossed once.
Find the probability of getting a multiple of 3.
Solution

### Example

3 blue marbles, 4 green marbles, 5 red marbles are in a jar.
If a marble is randomly picked from the jar,
P(blue marble) = ?
Solution

### Example

For the given spinner,
if you spin the arrow once,
P(X ≥ 4) = ?

Solution

## Probability: not A

### Formula

P(not A) = 1 - P(A)

### Example

Numbers: 1 ~ 10
If you randomly pick a number,
P(not a multiple of 3) = ?
Solution

## Probability: A and B

### Example

Numbers: 1 ~ 10
If you randomly pick a number,
P(odd and prime) = ?
Solution

## Probability: A or B

### Example

Numbers: 1 ~ 10
If you randomly pick a number,
P(odd or prime) = ?
Solution

### Formula

P(A or B) = P(A) + P(B) - P(A and B)

### Example

P(A) = 0.6, P(B) = 0.7, P(A and B) = 0.4
P(A or B) = ?
Solution

### Example

P(A) = 0.5, P(A and B) = 0.4, P(A or B) = 0.8
P(B) = ?
Solution

## Probability: Mutually Exclusive Events

### Formula

P(A or B) = P(A) + P(B)
Mutually exclusive events are the events
that don't happen together.
If A and B are mutually exclusive events,
P(A and B) = 0.
→ P(A or B) = P(A) + P(B)
Probability: A or B

### Example

Marbles are in a jar.
The probability of picking a blue marble is 0.3.
The probability of picking a green marble is 0.4.
If a marble is randomly picked from the jar,
P(blue marble or green marble) = ?
Solution

## Probability: Independent Events

### Formula

P(A and B) = P(A)P(B)
Independent events are the events
that do not affect each other.
So P(A) and P(B) do not affect each other.

### Example

A fair die and a coin is tossed once.
Solution

### Example

5 blue marbles, 4 green marbles, 3 red marbles are in a jar.
A marble is randomly picked from the jar and replaced.
This is repeated twice.
P(2 blue marbles) = ?
Solution

### Example

A, B: Independent events
P(A) = 47, P(A and B) = 17
P(A or B) = ?
Solution

## Probability: Dependent Events

### Formula

P(A and B) = P(A)P(B')
Dependent events are the events
that affect each other.
So P(A) affects P(B):
P(B) → P(B').

Probability: Independent Events

### Example

5 blue marbles, 4 green marbles, 3 red marbles are in a jar.
A marble is randomly picked from the jar and not replaced.
This is repeated twice.
P(2 blue marbles) = ?
Solution

## Conditional Probability

### Formula

P(B|A) = P(A and B)P(A)
P(B|A) means the probability of A and B
[B bar A] or [B given A].

### Example

80% of students like apple.
50% of students like both apple and banana.
If you choose a student who likes apple,
find the probability that the student also likes banana.
Solution

### Example

The probability of a student oversleeping is 4%.
The probability of the student oversleeping and getting late for school is 3%.
If the student woke up and realized that he overslept,
find the probability of the student getting late for school.
Solution