Probability of an event is the likelihood of the event.
An event (such as obtaining an even number in a single roll of a fair die) is made up of single values which we call outcomes. So, an outcome consist of only one of the possible values whereas an event is made up of one or more outcomes. In this tutorial, we cover the following topics:
Suppose there are m_1 ways to achieve task #1, m_2 ways to achieve task #2, m_3 ways to achieve a task #3, \cdots, m_k ways to achieve task #k, then there are
m_1m_2\cdots m_k
different ways to complete all k tasks.
Given that there are 5 items for appetizer, three items for the main meal, and 2 items for dessert. How many different meals are there? [A meal consists of an appetizer, an item from the main meal, and a dessert.]
Solution: 5\cdot 3\cdot 2 = 30 different meals
Two events A and B are mutually exclusive if the occurrence of one prevents the other.
For example, in a single roll of a fair die, let A = \text{getting an even number}, and B = \text{getting an odd number}. Then A and B are mutually exclusive as the occurrence of A precludes B, and vice-versa. Clearly, A\cap B = \phi, so that P(A|B) = P(B|A) = P(A\cap B) = 0, for mutually exclusive events A and B.
Suppose everyone in a company either gets policy A or B such that no one has both policies. Find the proportion of people who have policy A if a person with policy A is half as likely to get B.
Solution: In this case, P(A\cup B) = 1. Also, we have P(A\cup B) = P(A) + P(B) – P(A\cap B). Let P(A)= x. So, 1 = P(A) + P(B) – 0= x+2x – 0=3x \implies P(A) = x = \frac{1}{3}.