Table of Contents
- What is the Sample Space?
- An example of Finding the Sample Space
- Fundamental Counting Principle
- Example of Finding the total number of possible outcomes using the Fundamental Counting Principle
- What is a Compound Event ?
- Example of Finding the Probability of a Compound Event
- Frequently Asked Questions
In probability theory, the sample space is a collection of all possible outcomes or results of random experiments or events. You can find the sample space of two or more events using tables and tree diagrams. To find the total number of possible outcomes of two or more events, use the sample space or the fundamental counting principle.
Pick a type of bread and a type of sandwich randomly. Find the sample space. How many different sandwiches are possible?
Type of Bread – Multigrain
Type of sandwich/filling- Ham, Turkey, Steak, Chicken
Solution: Use a table to find the sample space.
Bread | Type of Sandwich/Filling | Outcomes |
|---|---|---|
Multigrain | Ham | Multigrain Ham |
Multigrain | Turkey | Multigrain Turkey |
Multigrain | Steak | Multigrain steak |
Multigrain | Chicken | Multigrain Chicken |
Since there are 4 different outcomes in the sample space, 4 different kinds of sandwiches are possible.
The fundamental counting principle can be extended to more than two events. An event M can have m different outcomes, while an event N can have n different outcomes. m × n is the total number of outcomes of event M followed by event N.
Find the total number of possible outcomes of flipping a coin and rolling a number cube.
Solution: We can gauge the total number of possible outcomes by using two methods.
Method 1: To find the sample space, use a table. Let H = heads and T = tails.
Cube | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
Coin head | 1H | 2H | 3H | 4H | 5H | 6H |
Coin tail | 1T | 2T | 3T | 4T | 5T | 6T |
Therefore, there are 12 possible outcomes.
Method 2: By using the fundamental counting principle, determine the number of different outcomes each event could have.
Event 1: Flipping a coin has 2 possible outcomes.
Event 2: Rolling a number cube has 6 possible outcomes.
Therefore, 6 × 2 or 12 possible outcomes exist.
A compound event is made up of two or more separate events. A compound event’s probability is the quotient of the number of favorable outcomes and the number of possible outcomes, just like a single event.
\(\frac{\text{The number of favorable outcomes}}{\text{Number of possible outcomes}}=\text{P (Event)} \)
Imagine tossing two fair coins. The result of the first event has no bearing on the outcome of the second. So, these are called independent events.
Imagine drawing a marble from a bag and, without replacing it, drawing a second marble from the bag. In this case, the outcome of the second event is determined by the first event’s outcome. These are called dependent events.
Example 1 : Consider that if a coin is tossed and a dice is rolled, following would be the possible outcomes. If so, what will be the probability of rolling a number greater than 2 and flipping tails.
Cube | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
Coin head | 1H | 2H | 3H | 4H | 5H | 6H |
Coin tail | 1T | 2T | 3T | 4T | 5T | 6T |
Solution: There are four favorable outcomes in the sample space for rolling a number greater than 2 and flipping tails: 3T, 4T, 5T and 6T.
\(\text{P (Event)} =\frac{\text{number of favorable outcomes}}{\text{number of possible outcomes}} \)
\(\text{P (greater than 2 and tails)}=\frac{4}{12} \)
\(=\frac{1}{3} \) or \( 33\% \)
The outcome of tossing the coin does not affect the outcome of tossing the cube. Hence they are independent events and not dependent events.
Example 2: You spin the spinner and flip a coin. Find the probability of the events – spinning a 2 and flipping tails; spinning a 7 and flipping heads; not spinning a 4 and flipping tails?
Solution: Probability of spinning a 2 and flipping tails \(=\frac{1}{2}\times\frac{1}{5}\)
\(=\frac{1}{10}\) or \(10\% \)
Probability of spinning a 7 and flipping heads \(=0\times\frac{1}{2}=0\)
Probability of not spinning a 4 and flipping tails \(=\frac{1}{2}\times\frac{4}{5}\)
\(=\frac{4}{10} \)
\(=\frac{2}{5} \) or \( 40\% \).
The fundamental counting principle comes in handy for finding the total number of possible outcomes of two or more events. We can use the sample space to find the number of outcomes of multiple events to get accurate results.
The set of all possible outcomes of one or more events can be understood as the sample space. You can find the sample space of two or more events by using tables and tree diagrams. The sample space is very important because it works as a substitute in calculating compound events.
A compound event consists of two or more events. Similar to a single event, the probability of a compound event is the quotient of the number of favorable outcomes and the number of possible outcomes.
The two types of compound events are:
- Dependent compound events.
- Independent compound events.
When the outcome of the first event does not affect the outcome of the second event, it is known as an independent event. Dependent events, on the contrary, are those where the outcome of the second event is determined by the outcome of the first event.