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

can you not take the square root of a negative number
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
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
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
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