Probability Operations

In this section, we'll talk about the various operations we can use with probability in order to make our calculations succinct and mathematical.

Operations

We'll go over several operation and examples, and leave you with some exercises.

Multiplication 
Rule

 

The multiplication rule is a rule we use for finding probabilities of independent events. Suppose we have 2 independent events A and B. We can say the probability of A to be P(A) and probability of B as P(B). If A and B are independent, then the probability of both A and B happening is P(A)*P(B).

Example:

Suppose we roll a dice 2 times. Find the probability of rolling a 5 and then rolling a 6.

Solution:

We can note that these are independent events: the outcome of one won't effect the other. Let A denote rolling a 5 and B denote rolling a 6. Then P(A)=1/6 and P(B)=1/6. Hence, using hte multiplication rule, the probability of both happening is P(A)*P(B)=1/36

A simpler way to understand the multiplication rule is as follows:

If P(A) and P(B) are fixed, then P(A and B) = P(A) * P(B)

Addition 
Rule

There is another rule in Probability that we call the Addition Rule. Suppose we have events A and B. Then P(A or B) = P(A) + P(B) - P(A and B). If A and B are mutually exclusive, then

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

Example:

Suppose there is a 5% chance of rain, and 30% chance that school will be closed. What is the Probability that either of those will occur?

Solution:

A is the event of rain happening, and B is the event of school closing. Then P(A)=0.05, and P(B)=0.3

So P(A or B) = 0.05 + 0.3 - P(A and B).

But we already know P(A and B) = P(A)*P(B) = 0.015. So P(A or B) = 0.35 - 0.015 = 0.335

One thing to note is that this formula is essentially identical to the Principle of Inclusion and Exclusion covered in Set Theory. If you don't know what this is, check out the site below

Replacement
Probability

Now we'll discuss the concept of replacement probability. An easy example to visualize is probaility of a deck of cards. Suppose we have a 52 card deck and we draw 2 cards randomly without replacement. What is the probability that these are the same cards?

 

You might think, as in the example with dice, that the answer is 1/52 * 1/52. But the key distinction here, is that the cards were drawn without placement. This means that after drawing the first card, we don't put it back in the deck. Finally, this means that the probability that they are the same must be 0.

Lets look at a more complicated example.

Problem:

Find the probability of drawing a king of hearts, and then drawing a queen of hearts?

Solution:

The probability of drawing a king of hearts is 1/52, but without replacement, the probability of finding a queen of hearts is now 1/51. Using our multiplication rule, we can deduce that the total probability is 1/52 * 1/51

Exercises:

1. Find the probability of getting 3 heads in a row, when flipping a fair coin

2. Find the probability of getting heads on a coin OR rolling a 6 on a dice

3. Find the probability of drawing 3 red cards in a row form a standard card deck

4. Find the probability getting 2 red cards in a row AND getting heads on a coin flip.