11.7 Probability (Page 3/18)

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

$\frac{7}{13}$

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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 \cup 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 $\frac{3}{6} = \frac{1}{2}$ and the probability of spinning a $d$ is $\frac{1}{6} .$ We can find the probability of spinning an orange or a $d$ simply by adding the two probabilities.

\begin{array}{l} P (E \cup^{​} F) = P (E) + P (F) \\ = \frac{1}{2} + \frac{1}{6} \\ = \frac{2}{3} \end{array}

The probability of spinning an orange or a $d$ is $\frac{2}{3} .$

A General Note

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 \cup F) = P (E) + P (F)

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.

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 $\frac{1}{4},$ and the probability of drawing a spade is also $\frac{1}{4},$ so the probability of drawing a heart or a spade is

\frac{1}{4} + \frac{1}{4} = \frac{1}{2}

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Try It

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

$\frac{2}{13}$

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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 $\frac{1}{9},$ the probability of the horse losing the race is simply

1 - \frac{1}{9} = \frac{8}{9}

A General Note

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.

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.

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

$1-1$	$1-2$	$1-3$	$1-4$	$1-5$	$1-6$
$2-1$	$2-2$	$2-3$	$2-4$	$2-5$	$2-6$
$3-1$	$3-2$	$3-3$	$3-4$	$3-5$	$3-6$
$4-1$	$4-2$	$4-3$	$4-4$	$4-5$	$4-6$
$5-1$	$5-2$	$5-3$	$5-4$	$5-5$	$5-6$
$6-1$	$6-2$	$6-3$	$6-4$	$6-5$	$6-6$

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
$\frac{3}{36} = \frac{1}{12}$
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.
$\begin{array}{l} P (E^{'}) = 1 - P (E) \\ = 1 - \frac{1}{12} \\ = \frac{11}{12} \end{array}$

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Questions & Answers

how does the planets on our solar system orbit

cheten Reply

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Issan Reply

Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.

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what is big bang theory?

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the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen

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I want to be an astronomer. That's my dream

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Who named the the whole galaxy?

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solar Univers

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what are the factors upon which the atmosphere is stratified

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is the big bang the sun

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bigbang is the beginning of the universe

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but thats just a theory

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nothing will happen, don't worry brother.

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what does comet means

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these are Rocky substances between mars and jupiter

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Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.

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how can we learn right and true ?

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why the moon is always appear in an elliptical shape

Gatjuol Reply

Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.

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what's see level?

lidiya Reply

Did you mean eye sight or sea level

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according to the theory of astronomers why the moon is always appear in an elliptical orbit?

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hi !!! I am new in astronomy.... I have so many questions in mind .... all of scientists of the word they just give opinion only. but they never think true or false ... i respect all of them... I believes whole universe depending on true ...থিউরি

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there many theory to born universe but what is the reality of big bang theory to born universe

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by big bang

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there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.

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from where on earth could u observe all the stars during the during the course of an year

Karuna Reply

I think it couldn't possible on earth

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in this time i don't Know

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is that so. the question was in the end of this chapter

Karuna

in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*). So it would just barely creep over the horizon.

Christopher

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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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