Probability of (A or B, Union)

Probability of (A or B, Union)

How to find the probability of (A or B, union): meaning, formula, examples, and their solutions.

Meaning

P(A or B) means the probability of either event A or event B happening.

P(A or B) means
the probability of
either event A or event B happening.

So, to find P(A or B),

find n(S),
total number of ways,

find n(A or B),
the number of ways
of either A or B happening,

then find P(A or B):
P(A or B) = n(A or B) / n(S).

Probability

Example 1

Numbers from 1 to 10 are given. If you randomly pick a number, find P(even or prime).

Numbers from 1 to 10 are given.

So there are 10 numbers
that can be picked.

So n(S) = 10.

Set the event A as
picking an even number.

The even numbers from 1 to 10 are
{2, 4, 6, 8, 10}.

And set the event B as
picking a prime number.

The prime numbers from 1 to 10 are
{2, 3, 5, 7}.

The numbers that are in either A or B are
{2, 3, 4, 5, 6, 7, 8, 10}.

So [A or B] is
{2, 3, 4, 5, 6, 7, 8, 10}.

[A or B] is {2, 3, 4, 5, 6, 7, 8, 10}.

So n(A or B) = 8.

n(S) = 10
n(A or B) = 8

So P(A or B) = 8/10.

Probability

Divide the numerator and the denominator by 2.

Then 4/5 is the answer.

Formula

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

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

When you add P(A) and P(B),
P(A and B) [green] is counted twice.

So P(A and B) is subtracted.

This formula is used
when both P(A or B) and P(A and B) are used together.

Probability of (A and B, Intersection)

Example 2

If P(A) = 0.6, P(B) = 0.7, and P(A and B) = 0.4, find P(A or B).

Write the given probabilities.

P(A) = 0.6
P(B) = 0.7
P(A and B) = 0.4

P(A) = 0.6
P(B) = 0.7
P(A and B) = 0.4

So P(A or B) = 0.6 + 0.7 - 0.4.

So P(A or B) = 0.9.

Example 3

If P(A) = 0.5, P(A and B) = 0.1, and P(A or B) = 0.8, find P(B).

Write the given probabilities.

P(A) = 0.5

P(B) is the one you want to find.
So set P(B) = x.

P(A and B) = 0.1
P(A or B) = 0.8

P(A) = 0.5
P(B) = x
P(A and B) = 0.1
P(A or B) = 0.8

So 0.5 + x - 0.1 = 0.8.

Solve the given equation.

Then x = 0.2.

So P(B) = 0.2.