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# Statistics

## Events

What is an event

Iprobability, 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. There are many different types of events that are applied in different situations. In this section we will be focusing on simple and compound events.

## Simple & Compound Events

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.

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

1. Determine whether these are simple or compound events:

2. a) Getting a number less than 2 or greater than 4 when spinning this spinner once.

1. b) Getting heads when a coin is tossed and getting a 3 when a six-sided number die is rolled.

2.

3. 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!)  