(A intersection B) U (A intersection B')

(A intersection B) U (A intersection B')
This is the [B]Universal Set U[/B].
Everything that isn't A and isn't B is everything else.

100 students were interviewed. 72 ate at A, 52 ate at B. How many ate at A and B?

P(A U B) = P(A) + P(B) - P(A intersection B)
100 = 72 + 52 - P(A intersection B)
P(A intersection B) = 72 + 52 - 100
P(A intersection B) = P(who ate at A and B) = 24

2 Lines Intersection

Free 2 Lines Intersection Calculator - Enter any 2 line equations, and the calculator will determine the following:

* Are the lines parallel?

* Are the lines perpendicular

* Do the lines intersect at some point, and if so, which point?

* Is the system of equations dependent, independent, or inconsistent

* Are the lines parallel?

* Are the lines perpendicular

* Do the lines intersect at some point, and if so, which point?

* Is the system of equations dependent, independent, or inconsistent

2x^2+4x < 3x+6

2x^2+4x < 3x+6
Subtract 3x from both sides:
2x^2 + x < 6
Subtract 6 from both sides
2x^2 + x - 6 < 0
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=2x%5E2%2Bx-6&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get:
x < 1.5 and x < -2
When we take the intersection of these, it's [B]x < 1.5[/B]

4 rectangular strips of wood, each 30 cm long and 3 cm wide, are arranged to form the outer section

4 rectangular strips of wood, each 30 cm long and 3 cm wide, are arranged to form the outer section of a picture frame. Determine the area inside the wooden frame.
Area inside forms a square, with a length of 30 - 3 - 3 = 24. We subtract 3 twice, because we account for 2 rectangular strips with a width of 3.
Area of a square is side * side. So we have 24 * 24 = [B]576cm^2[/B]

45 students, 12 taking spanish, 15 taking chemistry, 5 taking both spanish and chemistry. how many s

45 students, 12 taking spanish, 15 taking chemistry, 5 taking both spanish and chemistry. how many students are not taking either?
Let S be the number of students taking spanish and C be the number of students taking chemistry:
We have the following equation relating unions and intersections:
P(C U S) = P(C) + P(S) - P(C and S)
P(C U S) = 15 + 12 - 5
P(C U S) = 22
To get people that aren't taking either are, we have:
45 - P(C U S)
45 - 22
[B]23[/B]

A section of land measuring 3 & 3/6 acres is divided equally among 7 people. How many acres will eac

A section of land measuring 3 & 3/6 acres is divided equally among 7 people. How many acres will each person get?
We want 3&3/6 /7
[URL='https://www.mathcelebrity.com/fraction.php?frac1=3%263%2F6&frac2=7&pl=Divide']Using our fraction calculator[/URL], we get:
[B]1/2 acre per person[/B]

A spinner has 10 equally sized sections, 2 are gray 8 are blue. The spinner is spun twice. What is t

A spinner has 10 equally sized sections, 2 are gray 8 are blue. The spinner is spun twice. What is the probability that the first spin lands on blue and the second lands on gray?
P(blue) = Blue sections / Total Sections
P(blue) = 8/10
[URL='https://www.mathcelebrity.com/fraction.php?frac1=8%2F10&frac2=3%2F8&pl=Simplify']Reducing this using our simplified fraction calculator[/URL], we get:
P(blue) = 4/5
P(gray) = Gray sections / Total Sections
P(blue) = 2/10
[URL='https://www.mathcelebrity.com/fraction.php?frac1=2%2F10&frac2=3%2F8&pl=Simplify']Reducing this using our simplified fraction calculator[/URL], we get:
P(gray) = 1/5
We want the probability of blue,gray. Since each spin is independent, we multiply the two probabilities to get our answer:
P(blue, gray) = P(blue) * P(gray)
P(blue, gray) = 4/5 * 1/5
P(blue, gray) = [B]4/25[/B]

A spinner has 3 equal sections labelled A, B, C. A bag contains 3 marbles: 1 grey, 1 black, and 1 w

A spinner has 3 equal sections labelled A, B, C. A bag contains 3 marbles: 1 grey, 1 black, and 1 white. The pointer is spun and a marble is picked at random.
a) Use a tree diagram to list the possible outcomes.
[LIST=1]
[*][B]A, Grey[/B]
[*][B]A, Black[/B]
[*][B]A, White[/B]
[*][B]B, Grey[/B]
[*][B]B, Black[/B]
[*][B]B, White[/B]
[*][B]C, Grey[/B]
[*][B]C, Black[/B]
[*][B]C, White[/B]
[/LIST]
b) What is the probability of:
i) spinning A?
P(A) = Number of A sections on spinner / Total Sections
P(A) = [B]1/3[/B]
---------------------------------
ii) picking a grey marble?
P(A) = Number of grey marbles / Total Marbles
P(A) = [B]1/3[/B]
---------------------------------
iii) spinning A and picking a white marble?
Since they're independent events, we multiply to get:
P(A AND White) = P(A) * P(White)
P(A) was found in i) as 1/3
Find P(White):
P(White) = Number of white marbles / Total Marbles
P(White) = 1/3
[B][/B]
Therefore, we have:
P(A AND White) = 1/3 * 1/3
P(A AND White) = [B]1/9[/B]
---------------------------------
iv) spinning C and picking a pink marble?
Since they're independent events, we multiply to get:
P(C AND Pink) = P(C) * P(Pink)
Find P(C):
P(C) = Number of C sections on spinner / Total Sections
P(C) = 1/3
[B][/B]
Find P(Pink):
P(Pink) = Number of pink marbles / Total Marbles
P(Pink) = 0/3
[B][/B]
Therefore, we have:
P(C AND Pink) = 1/3 * 0
P(C AND Pink) = [B]0[/B]

A spinner has 6 equal sections, of which 2 are green. If you spin the spinner once, what is the prob

A spinner has 6 equal sections, of which 2 are green. If you spin the spinner once, what is the probability that it will land on a green section? Write your answer as a fraction or whole number.
P(green) = Total Green / Total spaces
P(green) = 2/6
We can simplify this fraction. So we [URL='https://www.mathcelebrity.com/fraction.php?frac1=2%2F6&frac2=3%2F8&pl=Simplify']type 2/6 into our search engine[/URL], choose Simplify, and we get:
P(green) = [B]1/3[/B]

A spinner is divided into 4 equal sections numbered 1 to 4. The theoretical probability of the spinn

A spinner is divided into 4 equal sections numbered 1 to 4. The theoretical probability of the spinner stopping on 3 is 25%. Which of the following is most likely the number of 3s spun in 10,000 spins?
We want Expected Value of s spins. Set up the expected value formula for any number 1-4
E(s) = 0.25 * n where n is the number of spins.
Using s = 3, n = 10,000, we have:
E(10,000) = 0.25 * 10,000
E(10,000) = [B]2,500[/B]

A woman walked for 5 hours, first along a level road, then up a hill, and then she turned around and

A woman walked for 5 hours, first along a level road, then up a hill, and then she turned around and walked back to the starting point along the same path. She walks 4mph on level ground, 3 mph uphill, and 6 mph downhill. Find the distance she walked.
Hint: Think about d = rt, which means that t = d/r. Think about each section of her walk, what is the distance and the rate. You know that the total time is 5 hours, so you know the sum of the times from each section must be 5.
Let Level distance = L and hill distance = H. Add the times it took for each section of the walk:
L/4 + H /3 + H/6 + L/4 = 5
The LCD of this is 12 from our [URL='http://www.mathcelebrity.com/gcflcm.php?num1=4&num2=3&num3=6&pl=LCM']LCD Calculator[/URL]
[U]Multiply each side through by our LCD of 12[/U]
3L + 4H + 2H + 3L = 60
[U]Combine like terms:[/U]
6L + 6H = 60
[U]Divide each side by 3:[/U]
2L + 2H = 20
The woman walked [B]20 miles[/B]

Dewey Decimal System Classification

Free Dewey Decimal System Classification Calculator - Given a 3 digit code, this will determine the class, division, and section of the library book using the Dewey Decimal System.

If A and B are independent events with P(A) = 0.2 and P(B) = 0.6, then P(A U B)=

If A and B are independent events with P(A) = 0.2 and P(B) = 0.6, then P(A U B)=?
We know the following formula for the probability of 2 events:
P(A U B) = P(A) + P(B) - P(A intersection B)
We're told A and B are independent, which makes P(A intersection B) = 0. So we're left with:
P(A U B) = P(A) + P(B) - P(A intersection B)
P(A U B) = 0.2 + 0.6 - 0
P(A U B) = [B]0.8[/B]

If n(A)=1200, n(B)=1250 and n(AintersectionB)=320, then n(AUB) is

If n(A)=1200, n(B)=1250 and n(AintersectionB)=320, then n(AUB) is
We know that:
n(AUB) = n(A) + n(B) - n(AintersectionB)
Plugging in our given numbers, we get:
n(AUB) = 1200 + 1250 - 320
n(AUB) = [B]2130[/B]

Let A and B be independent events with P(A) = 0.52 and P(B) = 0.62. a. Calculate P(A ? B).

Let A and B be independent events with P(A) = 0.52 and P(B) = 0.62. a. Calculate P(A ? B).
With independent events, the intersection probability is found by:
P(A ? B) = P(A) * P(B)
P(A ? B) = 0.52 * 0.62
P(A ? B) = [B]0.3224[/B]

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Refer to a bag containing 13 red balls numbered 1-13 and 5 green balls numbered 14-18. You choose a

Refer to a bag containing 13 red balls numbered 1-13 and 5 green balls numbered 14-18. You choose a ball at random.
a. What is the probability that you choose a red or even numbered ball?
b. What is the probability you choose a green ball or a ball numbered less than 5?
a. The phrase [I]or[/I] in probability means add. But we need to subtract even reds so we don't double count:
We have 18 total balls, so this is our denonminator for our fractions.
Red and Even balls are {2, 4, 6, 8, 10, 12}
Our probability is:
P(Red or Even) = P(Red) + P(Even) - P(Red and Even)
P(Red or Even) = 13/18 + 9/18 - 6/18
P(Red or Even) = 16/18
Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=16%2F18&frac2=3%2F8&pl=Simplify']Fraction Simplify Calculator[/URL], we have:
P(Red or Even) = [B]16/18[/B]
[B][/B]
b. The phrase [I]or[/I] in probability means add. But we need to subtract greens less than 5 so we don't double count:
We have 18 total balls, so this is our denonminator for our fractions.
Green and less than 5 does not exist, so we have no intersection
Our probability is:
P(Green or Less Than 5) = P(Green) + P(Less Than 5) - P(Green And Less Than 5)
P(Green or Less Than 5) = 5/18 + 4/18 - 0
P(Green or Less Than 5) = 9/18
Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=9%2F18&frac2=3%2F8&pl=Simplify']Fraction Simplify Calculator[/URL], we have:
P(Red or Even) = [B]1/2[/B]

Sections of a rail way are 66m in length. What is the length of 81 placed to end to end?

Sections of a rail way are 66m in length. What is the length of 81 placed to end to end?
We have 81 sections x 66 meters per section = [B]5,346[/B]

Set Notation

Free Set Notation Calculator - Given two number sets A and B, this determines the following:

* Union of A and B, denoted A U B

* Intersection of A and B, denoted A ∩ B

* Elements in A not in B, denoted A - B

* Elements in B not in A, denoted B - A

* Symmetric Difference A Δ B

* The Concatenation A · B

* The Cartesian Product A x B

* Cardinality of A = |A|

* Cardinality of B = |B|

* Jaccard Index J(A,B)

* Jaccard Distance J_{σ}(A,B)

* Dice's Coefficient

* If A is a subset of B

* If B is a subset of A

* Union of A and B, denoted A U B

* Intersection of A and B, denoted A ∩ B

* Elements in A not in B, denoted A - B

* Elements in B not in A, denoted B - A

* Symmetric Difference A Δ B

* The Concatenation A · B

* The Cartesian Product A x B

* Cardinality of A = |A|

* Cardinality of B = |B|

* Jaccard Index J(A,B)

* Jaccard Distance J

* Dice's Coefficient

* If A is a subset of B

* If B is a subset of A

The scale of a map shows that 1/2 inch is equal to 3/4 of a mile. How many inches on a map would be

The scale of a map shows that 1/2 inch is equal to 3/4 of a mile. How many inches on a map would be equal to 3 miles?
Set up a proportion of scale to actual distance
1/2 / 3/4 = x/3
4/3 = x/3
Cross multiply:
3x = 12
Divide each side by 3:
3x/3 = 12/3
x = [B]4 (1/2 inch sections) or 2 inches[/B]

There are 100 people in a sport centre. 67 people use the gym. 62 people use the swimming pool. 5

There are 100 people in a sport centre. 67 people use the gym. 62 people use the swimming pool. 56 people use the track. 38 people use the gym and the pool. 31 people use the pool and the track. 33 people use the gym and the track. 16 people use all three facilities. A person is selected at random. What is the probability that this person doesn't use any facility?
WE use the compound probability formula for 3 events:
[LIST=1]
[*]Gym use (G)
[*]Swimming pool use (S)
[*]Track (T)
[/LIST]
P(G U S U T) = P(G) + P(S) + P(T) - P(G Intersection S) - P(G Intersection T) - P(S Intersection T) + P(G Intersection S Intersection T)
[LIST]
[*]Note: U means Union (Or) and Intersection means (And)
[/LIST]
Plugging our numbers in:
P(G U S U T) = 67/100 + 62/100 + 56/100 - 38/100 - 31/100 - 33/100 + 16/100
P(G U S U T) = (67 + 62 + 56 - 38 - 31 - 33 + 16)/100
P(G U S U T) = 99/100 or 0.99
What this says is, the probability that somebody uses at any of the 3 facilities is 99/100.
The problem asks for none of the 3 facilities, or P(G U S U T)'
P(G U S U T)' = 1 - P(G U S U T)
P(G U S U T)' = 1 - 99/100
P(G U S U T)' = 100/100 - 99/100
P(G U S U T)' = [B]1/100 or 0.1[/B]

Venn Diagram

Free Venn Diagram Calculator - This lesson walks you through what a Venn diagram is, the Venn diagram for A union B, A intersection B, and A Complement.

Venn Diagram (2 circles)

Free Venn Diagram (2 circles) Calculator - Given two circles A and B with an intersection piece of C, this will calculate all relevant probabilities of the Venn Diagram.

What is a Perpendicular Bisector

Free What is a Perpendicular Bisector Calculator - This lesson walks you through what a perpendicular bisector is and the various properties of the segment it bisects and the angles formed by the bisection