Probability of events

Probability is a type of ratio where we compare how many times an outcome can occur compared to all possible outcomes.

P r o b a b i l i t y = T h e n u m b e r o f w a n t e d o u t c o m e s T h e n u m b e r o f p o s s i b l e o u t c o m e s

Example
What is the probability to get a 6 when you roll a die?
A die has 6 sides, 1 side contain the number 6 that give us 1 wanted outcome in 6 possible outcomes.

Independent events: Two events are independent when the outcome of the first event does not influence the outcome of the second event.
When we determine the probability of two independent events we multiply the probability of the first event by the probability of the second event.

P (X a n d Y) = P (X) \cdot P (Y)

To find the probability of an independent event we are using this rule:

Example
If one has three dice what is the probability of getting three 4s?
The probability of getting a 4 on one die is 1/6
The probability of getting 3 4s is:

P (4 a n d 4 a n d 4) = 1 6 \cdot 1 6 \cdot 1 6 = 1 216

When the outcome affects the second outcome, which is what we called dependent events.
Dependent events: Two events are dependent when the outcome of the first event influences the outcome of the second event. The probability of two dependent events is the product of the probability of X and the probability of Y AFTER X occurs.

P (X a n d Y) = P (X) \cdot P (Y a f t e r x)

Example
What is the probability for you to choose two red cards in a deck of cards?
A deck of cards has 26 black and 26 red cards. The probability of choosing a red card randomly is:

P (r e d) = 26 52 = 1 2

The probability of choosing a second red card from the deck is now:

P (r e d) = 25 51

The probability:

P (2 r e d) = 1 2 \cdot 25 51 = 25 102

Two events are mutually exclusive when two events cannot happen at the same time. The probability that one of the mutually exclusive events occur is the sum of their individual probabilities.

P (X o r Y) = P (X) + P (Y)

An example of two mutually exclusive events is a wheel of fortune. Let's say you win a bar of chocolate if you end up in a red or a pink field.

HERE A VIDEO OF EXPLANATIONS:

What is the probability that the wheel stops at red or pink?
P(red or pink)=P(red)+P(pink)

P (r e d) = 2 8 = 1 4

P (p i n k) = 1 8

P (r e d o r p i n k) = 1 8 + 2 8 = 3 8

Inclusive events are events that can happen at the same time. To find the probability of an inclusive event we first add the probabilities of the individual events and then subtract the probability of the two events happening at the same time.

P (X o r Y) = P (X) + P (Y) - P (X a n d Y)

Example
What is the probability of drawing a black card or a ten in a deck of cards?
There are 4 tens in a deck of cards P(10) = 4/52
There are 26 black cards P(black) = 26/52
There are 2 black tens P(black and 10) = 2/52

P (b l a c k o r t e n) = 4 52 + 26 52 - 2 52 = 30 52 - 2 52 = 28 52 = 7 13

Using the Fundamental Counting Principle to Determine the Sample Space

As we dive deeper into more complex probability problems, you may start wondering, "How can I figure out the total number of outcomes, also known as the sample space?"
We will use a formula known as the fundamental counting principle to easily determine the total outcomes for a given problem. First we are going to take a look at how the fundamental counting principle was derived, by drawing a tree diagram.

Example 1

We were able to determine the total number of possible outcomes (18) by drawing a tree diagram. However, this technique can be very time consuming.
The fundamental counting principle will allow us to take the same information and find the total outcomes using a simple calculation. Take a look.

Example 1 (continued)

As you can see, this is a much faster and more efficient way of determining the total outcomes for a situation.
Let's take a look at another example.

Example 2

I would not want to draw a tree diagram for Example 2! However, we were able to determine the total outcomes by using the fundamental counting principle.
Let's look at one more example and see how probability comes into play.

Example 3

Now it's your turn. No tree diagrams!

Practice Problem

Answer Key

Although you may think that drawing the tree diagrams is fun, it's much easier to use the formula, isn't it? I hope you had fun - now it's time to move on.
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Use These Examples of Probability To Guide You Through Calculating the Probability of Simple Events.

Probability is the chance or likelihood that an event will happen. It is the ratio of the number of ways an event can occur to the number of possible outcomes. We'll use the following model to help calculate the probability of simple events.

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Mathematics