The addition rule is used to calculate the probability of either or events. To calculate, we use the following formula:

The first part of the formula (P(A) + P(B)) makes complete sense. To get the union of the two events, we have to add together the area of the circles in the universe. But why the subtraction of P(A and B)? This is because when we add the two circles, we are adding the area of intersection twice, as shown in the following diagram:

See how both the red circles include the intersection of A and B? So, when we add them, we need to subtract just one of them to account for this, leaving us with our formula.

You will recall that we wanted the number of people who either had cancer or had a positive test result. If A is the event that someone has cancer, and B is that the test result was positive, we have the following:

P(A or B) = P(A) + P(B) - P(A and B) = .25 + .30 - .2 = .35

This is represented visually in the following diagram:

We say that two events are mutually exclusive if they cannot occur at the same time. This means that A∩B= ∅, or just that the intersection of the events is the empty set. When this happens, P(A ∩ B) = P(A and B) = 0.

If two events are mutually exclusive, then the following applies:

P(A ∪ B) = P(A or B)= P(A) + P(B) − P(A ∩ B) = P(A) + P(B)

This makes the addition rule much easier. Some examples of mutually exclusive events include the following:

None of these events can occur simultaneously.

The multiplication rule is used to calculate the probability of and events. To calculate P(A ∩ B) = P(A and B), we use the following formula:

P(A ∩ B) = P(A and B) = P(A) P(B|A)

Why do we use B|A instead of B? This is because it is possible that B depends on A. If this is the case, then just multiplying P(A) and P(B) does not give us the whole picture.

In our cancer trial example, let's find P(A and B). To do this, let's redefine A to be the event that the trial is positive, and B to be the person having cancer (because it doesn't matter what we call the events). The equation will be as follows:

P(A ∩ B) = P(A and B) = P(A) P(B|A) = .3 * .6666 = .2 = 20%

It's difficult to see the true necessity of using conditional probability, so let's try another, more difficult problem.

For example, of a randomly selected set of 10 people, 6 have iPhones and 4 have Androids. What is the probability that if I randomly select 2 people, they both will have iPhones? This example can be retold using event spaces, as follows:

I have the following two events:

So, basically, I want the following:

So, I can use my P(A and B) = P(A) P(B|A) formula.

P(A) is simple, right? People with iPhones are 6 out of 10, so, I have a 6/10 = 3/5 = 0.6 chance of A. This means P(A) = 0.6.

So, if I have a 0.6 chance of choosing someone with an iPhone, the probability of choosing two should just be 0.6 * 0.6, right?

But wait! We only have 9 people left to choose our second person from, because one was taken away. So, in our new transformed sample space, we have 9 people in total, 5 with iPhones and 4 with Androids, making P(B) = 5/9 = .555.

So, the probability of choosing two people with iPhones is 0.6 * 0.555 = 0.333 = 33%.

I have a 1/3 chance of choosing two people with iPhones out of 10. The conditional probability is very important in the multiplication rule as it can drastically alter your answer.

Two events are independent if one event does not affect the outcome of the other, that is P(B|A) = P(B) and P(A|B) = P(A).

If two events are independent, then the following applies:

P(A ∩ B) = P(A) P(B|A) = P(A) P(B)

Some examples of independent events are as follows:

None of these pairs of events affect each other.

The complement of A is the opposite or negation of A. If A is an event, Ā represents the complement of A. For example, if A is the event where someone has cancer, Ā is the event where someone is cancer free.

To calculate the probability of, Ā, use the following formula:

P(Ā) = 1 − P(A)

For example, when you throw two dice, what is the probability that you rolled higher than a 3?

Let A represent rolling higher than a 3.

Ā represents rolling a 3 or less.

P(A) = 1 − P(Ā)

P(A) = l - (P(2)+P(3))

= 1 - (2/36 + 2/36)

= 1 - (4/36)

= 32/36 = 8 / 9

= .89

For example, a start-up team has three investor meetings coming up. We will have the following probabilities:

What is the probability of them getting money from at least one meeting?

Let A be the team getting money from at least one investor, and, Ā, be the team not getting any money. P(A) can be calculated as follows:

P(A) = 1 − P(Ā)

To calculate P(Ā), we need to calculate the following:

P(Ā) = P(no money from investor 1 AND no money from investor 2 AND no money from investor 3)

Let's assume that these events are independent (they don't talk to each other):

P(Ā) = P(no money from investor 1) * P(no money from investor 2) * P(no money from investor 3) =

0.4 * 0.85 * 0.55 = 0.187

P(A) = 1 - 0.187 = 0.813 = 81%

So, the start-up has an 81% chance of getting money from at least one meeting!