Let S be a sample space and E and F be events associated with S . Suppose that Pr ( E ) = .5 , Pr ( F ) = .4 , and Pr ( E ∩ F ) = .1 . Calculate a. Pr ( E | F ) b. Pr ( F | E ) c. Pr ( E | F ′ ) d. Pr ( E ′ | F ′ ) .
Let S be a sample space and E and F be events associated with S . Suppose that Pr ( E ) = .5 , Pr ( F ) = .4 , and Pr ( E ∩ F ) = .1 . Calculate a. Pr ( E | F ) b. Pr ( F | E ) c. Pr ( E | F ′ ) d. Pr ( E ′ | F ′ ) .
Solution Summary: The author calculates the value of Pr(E|F) if the sample space is S and the events are E and F.
Let S be a sample space and E and F be events associated with S. Suppose that
Pr
(
E
)
=
.5
,
Pr
(
F
)
=
.4
,
and
Pr
(
E
∩
F
)
=
.1
. Calculate
a.
Pr
(
E
|
F
)
b.
Pr
(
F
|
E
)
c.
Pr
(
E
|
F
′
)
d.
Pr
(
E
′
|
F
′
)
.
Definition Definition For any random event or experiment, the set that is formed with all the possible outcomes is called a sample space. When any random event takes place that has multiple outcomes, the possible outcomes are grouped together in a set. The sample space can be anything, from a set of vectors to real numbers.
Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p-1)/2
multiple of n, i.e.
n mod p, 2n mod p, ...,
2
p-1
-n mod p.
Let T be the subset of S consisting of those residues which exceed p/2.
Find the set T, and hence compute the Legendre symbol (7|23).
The first 11 multiples of 7 reduced mod 23 are
7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8.
23
The set T is the subset of these residues exceeding
2°
So T = {12, 14, 17, 19, 21}.
By Gauss' lemma (Apostol Theorem 9.6),
(7|23) = (−1)|T| = (−1)5 = −1.
how come?
Shading a Venn diagram with 3 sets: Unions, intersections, and...
The Venn diagram shows sets A, B, C, and the universal set U.
Shade (CUA)' n B on the Venn diagram.
U
Explanation
Check
A-
B
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3. A different 7-Eleven has a bank of slurpee fountain heads. Their available flavors are as follows: Mountain
Dew, Mountain Dew Code Red, Grape, Pepsi and Mountain Dew Livewire. You fill five different cups full
with each type of flavor. How many different ways can you arrange the cups in a line if exactly two Mountain
Dew flavors are next to each other?
3.2.1
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Probability & Statistics (28 of 62) Basic Definitions and Symbols Summarized; Author: Michel van Biezen;https://www.youtube.com/watch?v=21V9WBJLAL8;License: Standard YouTube License, CC-BY
Introduction to Probability, Basic Overview - Sample Space, & Tree Diagrams; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=SkidyDQuupA;License: Standard YouTube License, CC-BY