# 11.7 Probability  (Page 2/18)

 Page 2 / 18

Construct a probability model for tossing a fair coin.

Outcome Probability
Roll of 1
Roll of 2
Roll of 3
Roll of 4
Roll of 5
Roll of 6

## Computing probabilities of equally likely outcomes

Let $\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ be a sample space for an experiment. When investigating probability, an event is any subset of $\text{\hspace{0.17em}}S.\text{\hspace{0.17em}}$ When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in $\text{\hspace{0.17em}}S.\text{\hspace{0.17em}}$ Suppose a number cube is rolled, and we are interested in finding the probability of the event “rolling a number less than or equal to 4.” There are 4 possible outcomes in the event and 6 possible outcomes in $\text{\hspace{0.17em}}S,\text{\hspace{0.17em}}$ so the probability of the event is $\text{\hspace{0.17em}}\frac{4}{6}=\frac{2}{3}.\text{\hspace{0.17em}}$

## Computing the probability of an event with equally likely outcomes

The probability of an event $E$ in an experiment with sample space $S$ with equally likely outcomes is given by

$\text{\hspace{0.17em}}E$ is a subset of $S,$ so it is always true that $0\le P\left(E\right)\le 1.\text{\hspace{0.17em}}$

## Computing the probability of an event with equally likely outcomes

A number cube is rolled. Find the probability of rolling an odd number.

The event “rolling an odd number” contains three outcomes. There are 6 equally likely outcomes in the sample space. Divide to find the probability of the event.

$\text{\hspace{0.17em}}P\left(E\right)=\frac{3}{6}=\frac{1}{2}\text{\hspace{0.17em}}$

A number cube is rolled. Find the probability of rolling a number greater than 2.

$\text{\hspace{0.17em}}\frac{2}{3}\text{\hspace{0.17em}}$

## Computing the probability of the union of two events

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     is the event that occurs if either or both events occur.

$\text{\hspace{0.17em}}P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)-P\left(E\cap F\right)\text{\hspace{0.17em}}$

Suppose the spinner in [link] is spun. We want to find the probability of spinning orange or spinning a $\text{\hspace{0.17em}}b.\text{\hspace{0.17em}}$

There are a total of 6 sections, and 3 of them are orange. So the probability of spinning orange is $\text{\hspace{0.17em}}\frac{3}{6}=\frac{1}{2}.\text{\hspace{0.17em}}$ There are a total of 6 sections, and 2 of them have a $\text{\hspace{0.17em}}b.\text{\hspace{0.17em}}$ So the probability of spinning a $\text{\hspace{0.17em}}b$ is $\frac{2}{6}=\frac{1}{3}.$ If we added these two probabilities, we would be counting the sector that is both orange and a $b$ twice. To find the probability of spinning an orange or a $b,$ we need to subtract the probability that the sector is both orange and has a $b.$

$\text{\hspace{0.17em}}\frac{1}{2}+\frac{1}{3}-\frac{1}{6}=\frac{2}{3}\text{\hspace{0.17em}}$

The probability of spinning orange or a $b\text{\hspace{0.17em}}$ is $\frac{2}{3}.$

## Probability of the union of two events

The probability of the union of two events $E$ and $F$ (written $\text{\hspace{0.17em}}E\cup F$ ) equals the sum of the probability of $E$ and the probability of $F$ minus the probability of $E$ and $F$ occurring together $\text{(}$ which is called the intersection of $E$ and $F$ and is written as $E\cap F$ ).

$\text{\hspace{0.17em}}P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)-P\left(E\cap F\right)\text{\hspace{0.17em}}$

## 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.

A standard deck contains an equal number of hearts, diamonds, clubs, and spades. So the probability of drawing a heart is $\text{\hspace{0.17em}}\frac{1}{4}.\text{\hspace{0.17em}}$ There are four 7s in a standard deck, and there are a total of 52 cards. So the probability of drawing a 7 is $\text{\hspace{0.17em}}\frac{1}{13}.$

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 $\text{\hspace{0.17em}}\frac{1}{52}.\text{\hspace{0.17em}}$ Substitute into the formula.

The probability of drawing a heart or a 7 is $\text{\hspace{0.17em}}\frac{4}{13}.$

#### Questions & Answers

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply

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