A die is thrown times and the. Random Experiments. Experimental Probability. Events in Probability. Empirical Probability. Coin Toss Probability. Probability of Tossing Two Coins.
Probability of Tossing Three Coins. Complimentary Events. Mutually Exclusive Events. Mutually Non-Exclusive Events. Conditional Probability. Theoretical Probability. Odds and Probability. Playing Cards Probability. When working with probability, a random action or series of actions is called a trial. An outcome is the result of a trial, and an event is a particular collection of outcomes.
Events are usually described using a common characteristic of the outcomes. Let's apply this language to see how the terms work in practice. Some games require rolling a die with six sides, numbered from 1 to 6. Dice is the plural of die. The chart below illustrates the use of trial, outcome, and event for such a game:. Examples of Events. There are 6 possible outcomes:. The event has no outcomes in it. Notice that a collection of outcomes is put in braces and separated by commas.
A simple event is an event with only one outcome. Rolling a 1 would be a simple event, because there is only one outcome that works—1! Rolling more than a 5 would also be a simple event, because the event includes only 6 as a valid outcome.
A compound event is an event with more than one outcome. For example, in rolling one six-sided die, rolling an even number could occur with one of three outcomes: 2, 4, and 6. When you roll a six-sided die many times, you should not expect any outcome to happen more often than another assuming that it is a fair die. The outcomes in a situation like this are said to be equally likely. Since each outcome in the die-rolling trial is equally likely, you would expect to get each outcome of the rolls.
That is, you'd expect of the rolls to be 1, of the rolls to be 2, of the rolls to be 3, and so on. A spinner is divided into four equal parts, each colored with a different color as shown below.
When this spinner is spun, the arrow points to one of the colors. Are the outcomes equally likely? A Yes, they are equally likely. B No, they are not equally likely. All the outcomes are equally likely. Each color provides a different outcome, and each color takes up of the circle. You would expect the arrow to point to each color of the time. Probability of Events. The probability of an event is how often it is expected to occur. It is the ratio of the size of the event space to the size of the sample space.
First, you need to determine the size of the sample space. The size of the sample space is the total number of possible outcomes. For example, when you roll 1 die, the sample space is 1, 2, 3, 4, 5, or 6. We can view the outcomes as two separate outcomes, that is, the outcome of rolling die number one and the outcome of rolling die number two.
This is somewhat more subtle than is first apparent. In this simple example, the outcomes of die number two have nothing to do with the outcomes of die number one. Here's a slightly more complicated example: how many ways are there to roll two dice so that the two dice don't match? That is, we rule out , , and so on. Here for each possible value on die number one, there are five possible values for die number two, but they are a different five values for each value on die number one.
This is often called the multiplication principle. This too can be proved by induction. Example 1. Note that we consider the dice to be distinguishable, that is, a roll of 6, 4, 1 is different than 4, 6, 1, because the first and second dice are different in the two rolls, even though the numbers as a set are the same. How many outcomes are possible? Definition 1. In example 1. The dice were distinguishable, or in a particular order: a first die, a second, and a third.
Now we want to count simply how many combinations of numbers there are, with 6, 4, 1 now counting as the same combination as 4, 6, 1. The list would contain many outcomes that we now wish to count as a single outcome; 6, 4, 1 and 4, 6, 1 would be on the list, but should not be counted separately. How many times will a single outcome appear on the list? You probably recognize these numbers: this is the beginning of Pascal's Triangle.
Each entry in Pascal's triangle is generated by adding two entries from the previous row: the one directly above, and the one above and to the left. Theorem 1. These values are the boundary ones in Pascal's Triangle. Many counting problems rely on the sort of reasoning we have seen. Here are a few variations on the theme. Measure ad performance. Select basic ads. Create a personalised ads profile. Select personalised ads.
Apply market research to generate audience insights. Measure content performance. Develop and improve products. List of Partners vendors. Share Flipboard Email. Courtney Taylor. Professor of Mathematics. Courtney K. Taylor, Ph. Updated February 02, Featured Video. Cite this Article Format.
Taylor, Courtney. Probabilities for Rolling Two Dice.
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