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Empirical probability based on survey data

A survey asks two questions of 300 students: Do you live on campus? Are you satisfied with the recreational facilities in the student center?Answers to the latter question were categorized “reasonably satisfied,” “unsatisfied,” or “no definite opinion.” Let C be the event “on campus;” O be the event “off campus;” S be the event “reasonably satisfied;” U be the event ”unsatisfied;” and N be the event “no definite opinion.” Data are shown in the following table.

Survey Data

Survey data
S U N
C 127 31 42
O 46 43 11

If an individual is selected on an equally likely basis from this group of 300, the probability of any of the events is taken to be the relative frequency of respondents ineach category corresponding to an event. There are 200 on campus members in the population, so P ( C ) = 200 / 300 and P ( O ) = 100 / 300 . The probability that a student selected is on campus and satisfied is taken to be P ( C S ) = 127 / 300 . The probability a student is either on campus and satisfied or off campus and not satisfied is

P ( C S O U ) = P ( C S ) + P ( O U ) = 127 / 300 + 43 / 300 = 170 / 300

If there is reason to believe that the population sampled is representative of the entire student body, then the same probabilities would be applied to any studentselected at random from the entire student body.

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It is fortunate that we do not have to declare a single position to be the “correct” viewpoint and interpretation.The formal model is consistent with any of the views set forth. We are free in any situation to make the interpretation most meaningful and natural to the problem at hand . It is not necessary to fit all problems into one conceptual mold; nor is itnecessary to change mathematical model each time a different point of view seems appropriate.

Probability and odds

Often we find it convenient to work with a ratio of probabilities. If A and B are events with positive probability the odds favoring A over B is the probability ratio P ( A ) / P ( B ) . If not otherwise specified, B is taken to be A c and we speak of the odds favoring A

O ( A ) = P ( A ) P ( A c ) = P ( A ) 1 - P ( A )

This expression may be solved algebraically to determine the probability from the odds

P ( A ) = O ( A ) 1 + O ( A )

In particular, if O ( A ) = a / b then P ( A ) = a / b 1 + a / b = a a + b .

O ( A ) = 0 . 7 / 0 . 3 = 7 / 3 . If the odds favoring A is 5/3, then P ( A ) = 5 / ( 5 + 3 ) = 5 / 8 .

Partitions and boolean combinations of events

The countable additivity property (P3) places a premium on appropriate partitioning of events.

Definition. A partition is a mutually exclusive class

{ A i : i J } such that Ω = i J A i

A partition of event A is a mutually exclusive class

{ A i : i J } such that A = i J A i

Remarks .

  • A partition is a mutually exclusive class of events such that one (and only one) must occur on each trial.
  • A partition of event A is a mutually exclusive class of events such that A occurs iff one (and only one) of the A i occurs.
  • A partition (no qualifier) is taken to be a partition of the sure event Ω .
  • If class { B i : ı J } is mutually exclusive and A B = i J B i , then the class { A B i : ı J } is a partition of A and A = i J A B i .

We may begin with a sequence { A 1 : 1 i } and determine a mutually exclusive (disjoint) sequence { B 1 : 1 i } as follows:

Questions & Answers

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_3_2_1
felecia
⅗ ⅔½
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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
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2+2x=
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Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
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4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
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Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
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Need help solving this problem (2/7)^-2
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x+2y-z=7
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-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
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Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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