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.
As you can see, with this formula, we will write the probability of an event as a fraction. The numerator (in red) is the number of chances and the denominator (in blue) is the set of all possible outcomes. This is also known as the sample space.
Example 1
Example 2
Hopefully these two examples have helped you to apply the formula in order to calculate the probability for any simple event.
Practice Problem
Answer Key
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Fundamental Counting Principle
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.
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.
Example 3
Answer Key
Probability of Independent Events
In this lesson, we will determine the probability of two events that are independent of one another. Let's first discuss what the term independent means in terms of probability.
Two events, A and B, are independent if the outcome of A does not affect the outcome of B.
In many cases, you will see the term, "With replacement". As we study a few probability problems, I will explain how "replacement" allows the events to be independent of each other.
Example 1
Example 1 is pretty easy to comprehend because we are finding the probability of two different events using two different tools. Let's see what happens when we use one tool, like a jar of marbles.
Example 2
This method for calculating the probability of independent events also works if you have more than 2 events occuring sequentially. Check out the practice problem below.
Practice Problems
Answer Key
Probability with Dependent Events
Have you been searching for probability help, specifically with dependent events? If you are new to Algebra-class.com or just starting a probability unit, you may want to take a look at the introductory probability lesson or the lesson on independent events. But... if you are ready to study dependent events, let's take a look at the definition.
Dependent Events
Two events, A and B, are dependent if the outcome of the first event does affect the outcome of the second event.
In many cases, the term "without replacement" will be used to signify dependent events.
Dependent Events are notated as: P(A,then B)
Example 1
Did you notice how the playing card was not replaced, so the outcomes and sample space were reduced for the second event? The second event is dependent on what happens on the first pick. Since this is theoretical probability and we don't know what would really happen on the first pick, we always assume that the first event happens as stated in the problem.
Example 2
We know that there's not a great chance that ALL 4 tires will be defective, but what are the chances that all four tires will NOT be defective?
Practice Problem
Answer Key
Theoretical Probability vs Experimental Probability
You've heard the terms, theoretical probability and experimental probability, but what do they mean? Are they in anyway related? This is what we are going to discover in this lesson.
If you've completed the lessons on independent and dependent probability, then you've already found the theoretical probability for numerous problems.
Theoretical Probability
Theoretical probability is the probability that is calculated using math formulas. This is the probability based on math theory.
Experimental Probability
Experimental probability is calculated when the actual situation or problem is performed as an experiment. In this case, you would perform the experiment, and use the actual results to determine the probability.
In order to accurately perform an experiment, you must:
· Identify what constitutes a "trial".
· Perform a minimum of 25 trials
· Set up an organizer (table or chart) to record your data.
Let's take a look at an example where we first calculate the theoretical probability, and then perform the experiment to determine the experimental probability. It will be interesting to compare the theoretical probability and the experimental probability. Do you think the two calculations will be close?
Example 1
This problem is from Example 1 in the independent events lesson. We calculated the theoretical probability to be 1/12 or 8.3%. Take a look:
Example 2
Probability of Compound Events
Thus far, we've studied several probability lessons. If you want to review a few of these lessons before studying compound events, check out the lessons on the fundamental counting principle, independent events, and dependent events.
If you are ready, let's move onto finding the probability of compound events.
Compound events can be further classified as mutually exclusive or mutually inclusive. The probability is calculated differently for each, so let's first take a look at mutually exclusive events.
Compound Events That Are Mutually Exclusive
When two events cannot happen at the same time, they are mutually exclusive events.
For example, you have a die and you are asked to find the probability of rolling a 1 or a 2. You know when you roll the die, only one of those numbers can appear, not both. Therefore, these events are mutually exclusive of each other.
Mutually Exclusive Events (Events that cannot happen at the same time)
P(A or B) = P(A) + P(B)
Take note: With this formula, you are adding the probabilites of each event, not multiplying.
Example 1
Mutually exclusive events are pretty straightforward. Now let's take a look at compound events that are inclusive.
When two events can occur at the same time, they are inclusive.
For example, let's take our example of rolling a regular 6-sided die. You are asked to find the probability of rolling a 2 or an even number. These events are inclusive because they can happen at the same time. A 2 is an even number, so this would satisfy both, but you could also roll a 4 or 6. Because 2 is a even number, these are inclusive events.
Inclusive Events (Events that CAN happen at the same time)
P(A or B) = P(A) + P(B) - P(A and B)
This is a little tricker, so let's take a look at example of inclusive events.
Example 2
I know that compound events can be confusing, but first you must determine if the events are exclusive or inclusive.
If the events are exclusive, then just add the probabilities of each individual event.
If the events are inclusive, you must remember to subtract the number outcomes that occur in both events.
Geometric Probability
Although you may have never formally heard the term "geometric probability", I bet you've often thought about it. Have you ever played darts? If so, have you thought about what your chances are of landing the dart on the bullseye? If so, you were thinking about geometric probability.
At first, geometric probability looks difficult. But, if you keep in mind the formula for basic probability, you will have no problems.
Formula for Probability = # of favorable outcomes / # of total outcomes
Example 1
Not too bad, if you know your geometry formulas. Now you know why they call it geometric probability.
Just to make sure you've got it, we'll take a look at one more example.
Example 2
Practice Problem
Answer Key