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Key equations
probability of an event with equally likely outcomes
$$P(E)=\frac{n(E)}{n(S)}$$
probability of the union of two events
$$P(E\cup F)=P(E)+P(F)-P(E\cap F)$$
probability of the union of mutually exclusive events
$$P(E\cup F)=P(E)+P(F)$$
probability of the complement of an event
$$P(E\text{'})=1-P(E)$$
Key concepts
Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain.
The probabilities in a probability model must sum to 1. See
[link] .
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 the sample space for the experiment. See
[link] .
To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. See
[link] .
To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. See
[link] .
The probability of the complement of an event is the difference between 1 and the probability that the event occurs. See
[link] .
In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces. See
[link] .
Section exercises
Verbal
What term is used to express the likelihood of an event occurring? Are there restrictions on its values? If so, what are they? If not, explain.
probability; The probability of an event is restricted to values between
$\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}$ inclusive of
$\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}1.\text{\hspace{0.17em}}$
The
union of two sets is defined as a set of elements that are present in at least one of the sets. How is this similar to the definition used for the
union of two events from a probability model? How is it different?
The probability of the
union of two events occurring is a number that describes the likelihood that at least one of the events from a probability model occurs. In both a union of sets
$\text{\hspace{0.17em}}A\text{}\text{and}B\text{\hspace{0.17em}}$ and a union of events
$\text{\hspace{0.17em}}A\text{and}B,\text{\hspace{0.17em}}$ the union includes either
$\text{\hspace{0.17em}}A\text{or}B\text{\hspace{0.17em}}$ or both. The difference is that a union of sets results in another set, while the union of events is a probability, so it is always a numerical value between
$\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}1.\text{\hspace{0.17em}}$
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
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
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
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
Brad
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
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
GREAT ANSWER THOUGH!!!
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