What is an event?
In probability, the set of outcomes from an experiment is known as an event. For instance, conducting an experiment on tossing a coin. The outcome in this experiment may be head or a tail - whatever takes place each time you toss the coin is the event.
Simple & Compound Events
A simple event is one that can only happen in one way - in other words, it has a single outcome. If we consider our previous example of tossing a coin: we get one outcome that is a head or a tail.
A compound event is more complex than a simple event, as it involves the probability of more than one outcome. Another way to view compound events is as a combination of two or more simple events.
Consider the probability of finding an even number less than 5. We have a combination of two simple events: finding an even number, and finding a number that is less than 5.
EXAMPLE
Determine whether these are simple or compound events:
a) Getting a number less than 2 or greater than 4 when spinning this spinner once.
b) Getting heads when a coin is tossed and getting a 3 when a six-sided number die is rolled.
See the video below for the solutions:
Probabilities for Simple and Compound Events
The probability of an event occurring requires two known variables: the number of times the event can occur, and the total number of possible outcomes. We use the following formula to calculate probability:
\[ Probability\ of\ event = \frac{Number\ of\ times\ it\ can\ occur}{Total\ number\ of\ possible\ outcomes} \]
Let’s try some problems!
1. Kyle works at a local music store. The store receives a shipment of new CDs of various genres in a box. In the shipment there are 10 country CDs, 5 rock CDs, 12 hip hop CDs, and 3 jazz CDs. What is the probability that the first CD Kyle chooses from the box will be country?
\( Step\ 1: \) How many Country CDs are there? number of times the event occurs
\( \Longrightarrow 10 \)
\( Step\ 2: \) How many CDs could Kyle choose from? total number of possible outcomes
\( \Longrightarrow 30 \)
\( Step\ 3: \) What is the probability that Kyle will choose a country CD first?
\( \Longrightarrow P(E) = \frac{10}{30} \) applying the probability formula
\( \Longrightarrow P(E) = \frac{1}{3} \) always reduce answer to lowest terms!
2. Kyle's store receives a new shipment of CDs in a box. In the shipment, there are 10 country CDs, 12 rock CDs, 5 hip hop CDs, and 3 jazz CDs.
What is the probability that Kyle will select a jazz CD from the box, and then, without replacing the CD, select a country CD?
This event consists of two simple events.
\( Step\ 1: \) What is the probability of selecting a jazz CD?
\( \Longrightarrow P(E_1) = \frac{3}{30} \)
\( Step\ 2: \) What is the probability of selecting a country CD without replacing the jazz CD?
\( \Longrightarrow \) What is our new total?
\( \Longrightarrow 29 \)
So the probability of selecting a country CD \( \Longrightarrow P(E_2) = \frac{10}{29} \)
\( Step\ 3: \) What is the probability of the first event taking place, followed by the second event?
\( \Longrightarrow P(E) = P(E_1) \times P(E_2) \)
\( \Longrightarrow\ \ \ \ \ \ \ \ \ = \frac{3}{30} \times \frac{10}{29} \)
\( \Longrightarrow\ \ \ \ \ \ \ \ \ = \frac{1}{29} \)
(Note - Final answer is determined by just doing multiplying two events when both are independent events. We will discuss independent and dependent events later on!)