# 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 $\text{\hspace{0.17em}}\frac{4}{13}.$

x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake