Probability Overview

The main three concepts that we need to understand from methods 1 & 2 are independent, mutually exclusive and conditional.

Independent Probability

Let’s look at independent probability first. If an event is independent, then whether or not another event B has of occuring has no effect on the probability of A occurring (e.g. the probability of rolling a six on a dice again, given that I have just rolled a six. Whether or not you have rolled a six, it does not affect your next outcome). This is expressed in the following way.

Pr(A|B)= Pr(A) This means that the probability of A, given that B has occurred is the same as the probability of A if B has not occurred. Now that we know that we can draw the following conclusion.

Pr(A B) = Pr(A) x Pr(B) Since the probability of A and B occurring aren’t related to each other, the chance of them both occurring is calculated simply by multiplying the chance of A by the chance of B.

Mutually Exclusive

Mutually exclusive is very simple: two events are mutually exclusive if they cannot both occur at the same time. This is expressed like this: Pr (A∩B)=0 What we are saying there is that the chance of both A and B occurring is equal to zero. I.e. that there is no chance of both A and B occurring.

Conditional Probability

Conditional probability is when the probability of an event is altered when it is known that another event has occurred. It is usually written as follows: Pr(A|B). For example, the probability of rain tomorrow changes based on the weather today. Pr(A|B)= $\frac{Pr(A\cap B)}{Pr(B)}$

Practice Question

Question 9b, i) from the 2014 VCAA exam 1.  As the probability of pleasant weather in the morning of April is 5 out of 8, it indicates that the probability of unpleasant weather is therefore, 3 out of 8 ( $1-\frac{3}{8}$). We can then draw a tree diagram for Sally to walk/not walk her dog Mack given that the weather is pleasant/unpleasant. To find the probability that Sally walked Mack on a particular morning, we times the probability of the pleasant morning, Pr(P) by the probability of Sally walking Mack given that the weather is pleasant, Pr (W|P). $\frac{5}{8} \times \frac{3}{4} = \frac{15}{32}$

We also needs to times the probability of the unpleasant morning, Pr(P’) by the probability of Sally walking Mack given that the weather is unpleasant, Pr(W|P’) $\frac{3}{8} \times \frac{1}{3} = \frac{3}{24}$

The final step is to add those two products together to find the final probability of Sally walking Mack on a particular morning in April. $\frac{15}{32} + \frac{3}{24} = \frac{19}{32}$