Lesson Elementary operations on sets help solving Probability problems
Algebra
->
Probability-and-statistics
-> Lesson Elementary operations on sets help solving Probability problems
Log On
Algebra: Probability and statistics
Section
Solvers
Solvers
Lessons
Lessons
Answers archive
Answers
Source code of 'Elementary operations on sets help solving Probability problems'
This Lesson (Elementary operations on sets help solving Probability problems)
was created by by
ikleyn(52780)
:
View Source
,
Show
About ikleyn
:
<H2>Elementary operations on sets help solving Probability problems</H2> <H3>Problem 1</H3>In a universal set U, A and B are subsets; n(A) = 24, n(B) = 60, and n(A ∩ B) = 10. Find n(A' ∩ B). <B>Solution</B> <pre> The set (A' ∩ B) is the set of those elements of B, that do not belong to A. So (A' ∩ B) is simply the difference of two subsets B and (A ∩ B) : (A' ∩ B) = B \ (A ∩ B), (*) where the symbol " \ " symbolizes subtraction of subsets. From (*), n(A' ∩ B) = n(B) - n(A ∩ B) = 60 - 10 = 50. <U>ANSWER</U> </pre> <H3>Problem 2</H3><pre> J, K, and L are events in sample space S. Pr(J)=0.35 Pr(K)=0.18 Pr(L)=0.24 Pr(J ∩ K)=0.11 Pr(J' ∩ L')=0.55 (*) Pr(K' ∩ L)=0.16 (**) What is Pr(J|K)? What is Pr(L|J)? What is Pr(K|L')?</pre>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ <B>Solution</B> 1. <U>What is Pr(J|K) ?</U> <pre> Pr(J|K) = by the definition = Pr(J ∩ K)/Pr(K) = {{{0.11/0.18}}} = {{{11/18}}}. </pre> 2. <U>What is Pr(L|J) ?</U> <pre> In the "given" part, find a line (*) related to L and J. This line is Pr(J' ∩ L')=0.55 From the elementary set theory, J' ∩ L' = (S\J) ∩ (S\L) = S \ (J U L). (Symbol " \ " means subtraction of sets.) You can prove it on your own using (or without using) Venn diagram for 2 subsets J and L in S. Also, from the basics of probability, Pr(J U L) = Pr(J) + Pr(L) - Pr((J ∩ L). These pieces are <U>THE KEY part of the solution</U>, which you <U>SHOULD and MUST understand</U> <U>when solving such problems</U> ! Therefore, 0.55 = Pr(J' ∩ L') = 1 - (Pr(J) + Pr(L)) + Pr((J ∩ L). which implies Pr(J ∩ L) = 0.55 + Pr(J) + Pr(L) - 1 = 0.55 + 0.35 + 0.24 - 1 = 0.14. Therefore, Pr(L|J) = by the definition = Pr(L∩J)/Pr(K) = {{{0.14/0.35}}} = {{{2/5}}}. </pre> 3. <U>What is Pr(K|L')?</U> <pre> By the definition, Pr(K|L') = P(K ∩ L')/P(L'). Pr(L') = 1 - Pr(L) = 1 - 0.24 = 0.76. What is P(K ∩ L') ? In the "given" part, find a line (**) related to K and L. This line is Pr(K' ∩ L)=0.16. The set (K' ∩ L) is "L without K", or, in other words, L\ (L ∩ K). Hence, Pr((K' ∩ L) = 0.16 = Pr(L) - Pr(L ∩ K), which gives Pr(L ∩ K) = Pr(L) - 0.16 = 0.024 - 0.16 = 0.08. The set (K ∩ L') is "K without L", or, in other words, K\ (L ∩ K). Hence, Pr(K ∩ L') = 0.18 - Pr(L ∩ K) = 0.18 - 0.08 = 0.10. Thus finally Pr(K|L') = {{{0.10/0.76}}} = {{{5/38}}}. </pre> <H3>Problem 3</H3>Let A and B be events with P(A) = 1/3, P(B) = 1/4 and P(AUB) = 1/2. Find the P(A ∩ B' ). <B>Solution</B> <pre> You may consider A and B as the subsets of the universal set U. Then (A ∩ B') = A \ (A ∩ B). ( Elements of A that are not in B ) The sign " \ " means subtraction of sets ) Therefore, you need first to find P(A ∩ B) and then subtract it from P(A). Step 1. P((A ∩ B) = P(A) + P(B) - P(AUB) = {{{1/3}}} + {{{1/4}}} - {{{1/2}}} = {{{(4+3-6)/12}}} = {{{1/12}}}. Step 2. P(A ∩ B') = P(A \ (A ∩ B)) = P(A) - P((A ∩ B) = {{{1/3}}} - {{{1/12}}} = {{{4/12}}} - {{{1/12}}} = {{{3/12}}} = {{{1/4}}}. <U>Answer</U>. P(A ∩ B') = {{{1/4}}} </pre> <H3>Problem 4</H3>Assume three events A, B, C are such that p(A)= 0.5, p(B)= 0.6, p(C)= 0.4, p(AnB)= 0.3, p(AnC)= 0.1, p(BnC)= 0.2, and p(AnBnC)= 0.05. Find (1) p(AnBnC') (2) p(AuBuC) (3) p(A'nBnC') <B>Solution</B> (1) <U>P(A n B n C')</U> <pre> To calculate P(A n B n C'), notice that (A n B n C') are those elements (events) of (A n B) that do not belong to C; in other words, (A n B n C') = (A n B) \ (A n B n C), (1) where the sign " \ " denotes subtraction of sets (of subsets). Then equality (1) implies that P(A n B n C') = P(A n B) - P(A n B n C) = (substitute the given data) = 0.3 - 0.05 = 0.25. <U>ANSWER</U> </pre> (2) <U>P(A U B U C)</U> <pre> Use the formula P(A U B U C) = P(A) + P(B) + P(C) - P(AnB) - P(AnC) - P(BnC) + P(A n B n C) (2) which is valid for any subsets A, B and C of the universal set U. This formula is very well known in elementary set theory. If you do not know its proof, here it is in few lines: Without my explanations, you understand well why I added P(A), P(B) and P(C) in the right side of the formula (2). But when I added these terms, I counted the intersections P(AnB), P(AnC) and P(BnC) twice - so I need to subtract these terms in the right side of the formula (2). But again, when I added the terms P(A), P(B) and P(C) in the right side of the formula (2), I counted the intersection P(A n B n C) thrice. When I later subtracted the terms P(AnB), P(AnC) and P(BnC), I fully compensate it, but now this intersection is not covered in the formula at all. Therefore I must add the term P(A n B n C), and this last step makes everything in equilibrium / (in balance). <U>The proof is completed</U>. Now, when the formula (2) is proved, simply substitute the given input data in it. You will get P(A U B U C) = 0.5 + 0.6 + 0.4 - 0.3 - 0.1 - 0.2 + 0.05 = 0.95 <U>ANSWER</U> </pre> (3) <U>P(A' n B n C')</U> <pre> To calculate P(A' n B n C'), notice that (A' n B n C') are those elements (events) of B that do belong NEITHER A NOR C; in other words, (A' n B n C') = (B \ (A n B) \ (C n B)) (3) Then equality (2) implies that P(A' n B n C') = P(B \ (A n B) \ (C n B)) = (P(B) - P(AnB)) + (P(B) - P(CnB)) + P(A n B n C) (4) Notice that in the right side of the formula (4), I subtracted the probability P(AnB) from P(B), then subtracted P(CnB) from P(B), so I subtracted the probability P(AnC) of the intersection (AnC) twice; therefore, I should add P(A n B n C) to restore the equilibrium. Substitute the given data into the formula (4), you get <U>the ANSWER</U> P(A' n B n C') = (0.6 - 0.3 - 0.2) + 0.05 = 0.15. (5) </pre> My other lessons on <B>Probability</B> in this site are <TABLE> <TR> <TD> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Simple-and-simplest-probability-problems.lesson>Simple and simplest probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Solving-probability-problems-using-complementary-probability.lesson>Solving probability problems using complementary probability</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Elementary-Probability-problems-related-to-Combinations.lesson>Elementary Probability problems related to combinations</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-True-False-test.lesson>A True/False test</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-multiple-choice-answers-test.lesson>A multiple choice answers test</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Coinciding-birthdays.lesson>Coinciding birthdays</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-shipment-containing-fair-and-defective-alarm-clocks-.lesson>A shipment containing fair and defective alarm clocks</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/People-in-a-room-write-down-integer-numbers-at-random.lesson>People in a room write down integer numbers at random</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-drawer-contains-a-mixture-of-socks.lesson>A drawer contains a mixture of socks</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Students-studying-foreign-languages.lesson>Students studying foreign languages</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-for-a-computer-to-be-damaged-by-viruses.lesson>Probability for a computer to be damaged by viruses</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Conditional-probability-problems.lesson>Conditional probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Using-sample-space-to-solve-Probability-problems.lesson>Using sample space to solve Probability problems</A> </TD> <TD> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Typical-probability-problems-from-the-archive.lesson>Typical probability problems from the archive</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Geometric-probability-problems.lesson>Geometric probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Advanced-probability-problems-from-the-archive.lesson>Advanced probability problems from the archive</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Challenging-probability-problems.lesson>Challenging probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Selected-probability-problems-from-the-archive.lesson>Selected probability problems from the archive</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Experimental-probability-problems.lesson>Experimental probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Rolling-a-pair-of-fair-dice.lesson>Rolling a pair of fair dice</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Ten-pairs-of-shoes-are-kept-in-a-rack.lesson>Ten pairs of shoes are kept in a rack</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-diplomat-and-spies.lesson>A diplomat and spies</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Independent-and-mutually-exclusive-events.lesson>Independent and mutually exclusive events</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Unusual-probability-problems.lesson>Unusual probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-problem-for-the-Day-of-April-1.lesson>Probability problem for the Day of April, 1</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/OVERVIEW-of-lessons-on-Probability.lesson>OVERVIEW of lessons on Probability</A> </TD> </TR> </TABLE> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
