Probability for Multiple Events
Learning Objectives
- Find the probability of a union of two events
- Find the probability of two events that share no common outcomes
- Find the probability that an event will not happen
- Find the number of events in a sample space that that includes many choices
We are often interested in finding the probability that one of multiple events occurs. Suppose we are playing a card game, and we will win if the next card drawn is either a heart or a king. We would be interested in finding the probability of the next card being a heart or a king. The union of two events [latex]E\text{ and }F,\text{written }E\cup F[/latex], is the event that occurs if either or both events occur.
[latex]P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)-P\left(E\cap F\right)[/latex]
Suppose the spinner below is spun. We want to find the probability of spinning orange or spinning a [latex]b[/latex].
There are a total of 6 sections, and 3 of them are orange. So the probability of spinning orange is [latex]\frac{3}{6}=\frac{1}{2}[/latex]. There are a total of 6 sections, and 2 of them have a [latex]b[/latex]. So the probability of spinning a [latex]b[/latex] is [latex]\frac{2}{6}=\frac{1}{3}[/latex]. If we added these two probabilities, we would be counting the sector that is both orange and a [latex]b[/latex] twice. To find the probability of spinning an orange or a [latex]b[/latex], we need to subtract the probability that the sector is both orange and has a [latex]b[/latex].
[latex]\frac{1}{2}+\frac{1}{3}-\frac{1}{6}=\frac{2}{3}[/latex]
The probability of spinning orange or a [latex]b[/latex] is [latex]\frac{2}{3}[/latex].A General Note: Probability of the Union of Two Events
The probability of the union of two events [latex]E[/latex] and [latex]F[/latex] (written [latex]E\cup F[/latex] ) equals the sum of the probability of [latex]E[/latex] and the probability of [latex]F[/latex] minus the probability of [latex]E[/latex] and [latex]F[/latex] occurring together [latex]\text{(}[/latex] which is called the intersection of [latex]E[/latex] and [latex]F[/latex] and is written as [latex]E\cap F[/latex] ). [latex-display]P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)-P\left(E\cap F\right)[/latex-display]Example: Computing the Probability of the Union of Two Events
A card is drawn from a standard deck. Find the probability of drawing a heart or a 7.Answer: A standard deck contains an equal number of hearts, diamonds, clubs, and spades. So the probability of drawing a heart is [latex]\frac{1}{4}[/latex]. There are four 7s in a standard deck, and there are a total of 52 cards. So the probability of drawing a 7 is [latex]\frac{1}{13}[/latex]. The only card in the deck that is both a heart and a 7 is the 7 of hearts, so the probability of drawing both a heart and a 7 is [latex]\frac{1}{52}[/latex]. Substitute [latex]P\left(H\right)=\frac{1}{4}, P\left(7\right)=\frac{1}{13}, \text{and} P\left(H\cap 7\right)=\frac{1}{52}[/latex] into the formula.
[latex]\begin{array}{l}P\left(E{\cup }^{\text{ }}F\right)=P\left(E\right)+P\left(F\right)-P\left(E{\cap }^{\text{ }}F\right)\hfill \\ \text{ }=\frac{1}{4}+\frac{1}{13}-\frac{1}{52}\hfill \\ \text{ }=\frac{4}{13}\hfill \end{array}[/latex]
The probability of drawing a heart or a 7 is [latex]\frac{4}{13}[/latex].Try It
A card is drawn from a standard deck. Find the probability of drawing a red card or an ace.Answer: [latex]\frac{7}{13}[/latex]
Computing the Probability of Mutually Exclusive Events
Suppose the spinner from earlier is spun again, but this time we are interested in the probability of spinning an orange or a [latex]d[/latex]. There are no sectors that are both orange and contain a [latex]d[/latex], so these two events have no outcomes in common. Events are said to be mutually exclusive events when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is[latex]P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)[/latex]
Notice that with mutually exclusive events, the intersection of [latex]E[/latex] and [latex]F[/latex] is the empty set. The probability of spinning an orange is [latex]\frac{3}{6}=\frac{1}{2}[/latex] and the probability of spinning a [latex]d[/latex] is [latex]\frac{1}{6}[/latex]. We can find the probability of spinning an orange or a [latex]d[/latex] simply by adding the two probabilities.[latex]\begin{array}{l}P\left(E{\cup }^{\text{ }}F\right)=P\left(E\right)+P\left(F\right)\hfill \\ \text{ }=\frac{1}{2}+\frac{1}{6}\hfill \\ \text{ }=\frac{2}{3}\hfill \end{array}[/latex]
The probability of spinning an orange or a [latex]d[/latex] is [latex]\frac{2}{3}[/latex].A General Note: Probability of the Union of Mutually Exclusive Events
The probability of the union of two mutually exclusive events [latex]E\text{and}F[/latex] is given by[latex]P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)[/latex]
How To: Given a set of events, compute the probability of the union of mutually exclusive events.
- Determine the total number of outcomes for the first event.
- Find the probability of the first event.
- Determine the total number of outcomes for the second event.
- Find the probability of the second event.
- Add the probabilities.
Example: Computing the Probability of the Union of Mutually Exclusive Events
A card is drawn from a standard deck. Find the probability of drawing a heart or a spade.Answer: The events "drawing a heart" and "drawing a spade" are mutually exclusive because they cannot occur at the same time. The probability of drawing a heart is [latex]\frac{1}{4}[/latex], and the probability of drawing a spade is also [latex]\frac{1}{4}[/latex], so the probability of drawing a heart or a spade is
[latex]\frac{1}{4}+\frac{1}{4}=\frac{1}{2}[/latex]
Try It
A card is drawn from a standard deck. Find the probability of drawing an ace or a king.Answer: [latex]\frac{2}{13}[/latex]
Find the Probability That an Even Will Not Happen
We have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding the probability that an event will not happen. The complement of an event [latex]E[/latex], denoted [latex]{E}^{\prime }[/latex], is the set of outcomes in the sample space that are not in [latex]E[/latex]. For example, suppose we are interested in the probability that a horse will lose a race. If event [latex]W[/latex] is the horse winning the race, then the complement of event [latex]W[/latex] is the horse losing the race. To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1.[latex]P\left({E}^{\prime }\right)=1-P\left(E\right)[/latex]
The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is [latex]\frac{1}{9}[/latex], the probability of the horse losing the race is simply[latex]1-\frac{1}{9}=\frac{8}{9}[/latex]
A General Note: The Complement Rule
The probability that the complement of an event will occur is given by [latex-display]P\left({E}^{\prime }\right)=1-P\left(E\right)[/latex-display]Example: Using the Complement Rule to Calculate Probabilities
Two six-sided number cubes are rolled.- Find the probability that the sum of the numbers rolled is less than or equal to 3.
- Find the probability that the sum of the numbers rolled is greater than 3.
Answer: The first step is to identify the sample space, which consists of all the possible outcomes. There are two number cubes, and each number cube has six possible outcomes. Using the Multiplication Principle, we find that there are [latex]6\times 6[/latex], or [latex]\text{ 36 }[/latex] total possible outcomes. So, for example, 1-1 represents a 1 rolled on each number cube.
| [latex]\text{1 - 1}[/latex] | [latex]\text{1 - 2}[/latex] | [latex]\text{1 - 3}[/latex] | [latex]\text{1 - 4}[/latex] | [latex]\text{1 - 5}[/latex] | [latex]\text{1 - 6}[/latex] |
| [latex]\text{2 - 1}[/latex] | [latex]\text{2 - 2}[/latex] | [latex]\text{2 - 3}[/latex] | [latex]\text{}[/latex] [latex]\text{2 - 4}[/latex] | [latex]\text{2 - 5}[/latex] | [latex]\text{2 - 6}[/latex] |
| [latex]\text{3 - 1}[/latex] | [latex]\text{3 - 2}[/latex] | [latex]\text{3 - 3}[/latex] | [latex]\text{3 - 4}[/latex] | [latex]\text{3 - 5}[/latex] | [latex]\text{3 - 6}[/latex] |
| [latex]\text{4 - 1}[/latex] | [latex]\text{4 - 2}[/latex] | [latex]\text{4 - 3}[/latex] | [latex]\text{4 - 4}[/latex] | [latex]\text{4 - 5}[/latex] | [latex]\text{4 - 6}[/latex] |
| [latex]\text{5 - 1}[/latex] | [latex]\text{5 - 2}[/latex] | [latex]\text{5 - 3}[/latex] | [latex]\text{5 - 4}[/latex] | [latex]\text{5 - 5}[/latex] | [latex]\text{5 - 6}[/latex] |
| [latex]\text{6 - 1}[/latex] | [latex]\text{6 - 2}[/latex] | [latex]\text{6 - 3}[/latex] | [latex]\text{6 - 4}[/latex] | [latex]\text{6 - 5}[/latex] | [latex]\text{6 - 6}[/latex] |
- We need to count the number of ways to roll a sum of 3 or less. These would include the following outcomes: 1-1, 1-2, and 2-1. So there are only three ways to roll a sum of 3 or less. The probability is
[latex]\frac{3}{36}=\frac{1}{12}[/latex]
- Rather than listing all the possibilities, we can use the Complement Rule. Because we have already found the probability of the complement of this event, we can simply subtract that probability from 1 to find the probability that the sum of the numbers rolled is greater than 3.
[latex]\begin{array}{l}P\left({E}^{\prime }\right)=1-P\left(E\right)\hfill \\ \text{ }=1-\frac{1}{12}\hfill \\ \text{ }=\frac{11}{12}\hfill \end{array}[/latex]
Try It
Two number cubes are rolled. Use the Complement Rule to find the probability that the sum is less than 10.Answer: [latex]\frac{5}{6}[/latex]
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- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
