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A card is drawn from a standard deck. Find the probability of drawing a red card or an ace.

7 13

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Computing the probability of mutually exclusive events

Suppose the spinner in [link] is spun again, but this time we are interested in the probability of spinning an orange or a d . There are no sectors that are both orange and contain a d , 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

P ( E F ) = P ( E ) + P ( F )

Notice that with mutually exclusive events, the intersection of E and F is the empty set. The probability of spinning an orange is 3 6 = 1 2 and the probability of spinning a d is 1 6 . We can find the probability of spinning an orange or a d simply by adding the two probabilities.

P ( E F ) = P ( E ) + P ( F )                 = 1 2 + 1 6                 = 2 3

The probability of spinning an orange or a d is 2 3 .

Probability of the union of mutually exclusive events

The probability of the union of two mutually exclusive events E and F is given by

P ( E F ) = P ( E ) + P ( F )

Given a set of events, compute the probability of the union of mutually exclusive events.

  1. Determine the total number of outcomes for the first event.
  2. Find the probability of the first event.
  3. Determine the total number of outcomes for the second event.
  4. Find the probability of the second event.
  5. Add the probabilities.

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.

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 1 4 , and the probability of drawing a spade is also 1 4 , so the probability of drawing a heart or a spade is

1 4 + 1 4 = 1 2
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A card is drawn from a standard deck. Find the probability of drawing an ace or a king.

2 13

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Using the complement rule to compute probabilities

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 E , denoted E , is the set of outcomes in the sample space that are not in E . For example, suppose we are interested in the probability that a horse will lose a race. If event W is the horse winning the race, then the complement of event W 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.

P ( E ) = 1 P ( E )

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 1 9 , the probability of the horse losing the race is simply

1 1 9 = 8 9

The complement rule

The probability that the complement of an event    will occur is given by

P ( E ) = 1 P ( E )

Using the complement rule to calculate probabilities

Two six-sided number cubes are rolled.

  1. Find the probability that the sum of the numbers rolled is less than or equal to 3.
  2. Find the probability that the sum of the numbers rolled is greater than 3.

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 6 × 6 , or  36  total possible outcomes. So, for example, 1-1 represents a 1 rolled on each number cube.

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
  1. 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
    3 36 = 1 12
  2. 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.
    P ( E ) = 1 P ( E )           = 1 1 12           = 11 12
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Questions & Answers

a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
Practice Key Terms 9

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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