What is Probability? Probability is a measure of the likelihood that an event will happen. In mathematical terms, probability is defined as the favorable outcomes over all the possible outcomes. When discussing probability the possible outcomes are the possible results of a test. For example, when rolling a die the possible outcomes are 1,2,3,4,5 and 6.
How to use rolling a die in probability: You can use rolling a die in probability to test outcomes of a specific event.
An event is the set of the actual outcomes of a test. It is a subset of the sample space.
For example, when rolling a die, an event can be rolling a prime number. The die would need to land on a 2,3, or 5. There are many different events that can occur related to rolling a die.
A sample space is all the possible outcomes of a test.
When rolling a die, the sample space is 1,2,3,4,5,6. There are 6 possible outcomes.
The number of events and possible outcomes will increase as you add more dice to the test.
For example, when you roll 2 dice, the sample space jumps from 6 possible outcomes to 36. They are:
{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
The probability of an event happening lies between 0 and 1. The probability will be 0, if there is no possibility that the event will occur. The probability will be 1, if it is certain that the event will happen. Anything that falls between 0 and 1 will show how likely or unlikely it is that the event will happen.
When rolling a die, the probability of rolling a 7 would be 0, because 7 does not exist on a die.If asked the probability of rolling any number between 1 and 6, the probability would be 1 because you are certain to roll one of those numbers.
To find the probability of an event, you can use the equation:
P= # of favorable outcomes/ total # of possible outcomes
There are two types of probability. They are theoretical probability and experimental probability.
Theoretical Probability:
Theoretical Probability is calculated by what is mathematically expected to happen.
For example when rolling a 6 sided die, the theoretical probability of rolling a 1 or 3 is 2/6 or ⅓.
In this example, there are only 2 chances of rolling a 1 or a 3 because each number is only found once on a standard die.
The favorable outcome will equal 2.
The total number of possible outcomes will equal 6 because the die can land on a 1,2,3,4,5, or 6.
When we put this in the equation P= # of favorable outcomes/ total # of possible outcomes, it will be P=2/6.
The probability of rolling a 1 or a 3 will be 2/6. When simplified P = 1/3.
Experimental Probability:
Experimental Probability is based on the outcomes of repeated tests.
For example, if you were to roll a die 400 times and the die landed on the number three 210 times, then the experimental probability would be 210/400 or 21/40.
In this example, the probability of the die landing on a 3 was tested 400 times. The number of favorable outcomes was 210 and the total number of tests was 400,
Using the equation P = # of favorable outcomes/ total number of outcomes we find that the experimental probability is 210/400. When simplified P = 21/40.
Sample Math Problems
Question
Two dice are rolled simultaneously. What is the probability of rolling a sum of 9 or 8?
Solution
First, find all the possible outcomes of rolling two dice. They are:
{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
There are 36 total possible outcomes.
Now we need to find out the favorable outcomes for rolling a sum of 9 or 8.
They are:
{(2,6),(3,5),(3,6),(4,4),(4,5),(5,3),(5,4),(6,2),(6,3)}
There are 9 favorable outcomes.
Now use the formula P=favorable outcomes/ total possible outcomes.
P = 9/36.
Lastly, simplify.
P = ¼
Question
Michael rolls two unbiased dice. He wants odd numbers on both the dice. Find the number of favorable outcomes of rolling odd numbers on both the dice.
Solution
First look at the total possibilities of rolling two dice.
{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
From these only pick the possibilities that have two odd numbers. These will be the favorable outcomes.
They are :
{(1,1),(1,3),(1,5),(3,1),(3,3),(3,5),(5,1),(5,3),(5,5)}
There are 9 favorable outcomes.
Question
If a die is rolled 400 times and a 4 is rolled 311 times, what is the experimental probability of rolling a 4?
