Question

MAT150-Introduction to Statistics Writing Assignment # 3 Topic: Probability Write a brief essay in which you summarize the main points Please include formulas if needed. Discuss the concepts of mutually exclusive events and independent events. List several examples of each type of event from everyday life You need to answer only two questions. One of them must be the first question. Then you can choose between the second or third question. 1. If A andB are mutually exclusive events, does it follow that An and B cannot be independent events ? Give an example to demonstrate your answer. For example, discuss an election where only one person can win. Let A be the event that party A's candidate wins, let B be the event that party B's candidate wins, Does the outcome of one event determine the outcome of the other event ? Are A and B mutually exclusive events? 2. Discuss conditions under which PIA and B) Under what conditions is this not true ? P(A) P(B) is true. 3. Discuss conditions under which P(A or B) Under which conditions is this not true ? P(A) + P(B) is true. MacBook Pro
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Answer 1

$Q1.~Mutually~Exclusive~Events:~Events~are~said~to~be~mutually~exclusive~when\\ no~two~of~them~can~occur~simultaneously.~\\ Independent~events:~Two~events~A~and~B~are~independent~if\\ ~P(A|B)=P(A)~and~P(B|A)=P(B).\\ Consider~families~with~3~children.~Sample~space=\{(bbb),(bbg),(bgb),(bgg),(gbb),(gbg),(ggb),(ggg)\},\\ where~"b"~stands~for~a~boy~and~"g"~for~a~girl.~Assume~the~cases~are~all~equally~likely.~Let~A~be~an~event~that\\ the~family~has~children~of~both~sexes,~B~be~an~event~that~there~is~at~most~one~girl,\\ C~be~event~that~there~is~at~least~two~girls.\\ Then~A=\{(bbg),(bgb),(bgg),(gbb),(gbg),(ggb)\};~B=\{(bbb),(bbg),(bgb),(gbb)\};\\ ~C=\{(bgg),(gbg),(ggb),(ggg)\}. \\ Then~we~see~that~P(A|B)=3/4=P(A)~and~P(B|A)=3/6=1/2=P(B),\\ hence~A~and~B~are~independent.~Since~P(B\cap C)=0~so~B~and~C~are~mutually~exclusive~events.\\ From~this~example~we~see~that~\\ P(B\cap C)=0\neq P(B)P(C)=(1/4)*(1/4)=1/16.\\$$No,~the~outcome~of~one~event~does~not~determine~the~outcome~of~the~other~event. \\ ~Hence~A~and~B~are~independent.\\ 2.~When~A~and~B~are~independent~then~P(A|B)=P(A)\\ ~hence~P(A\cap B)=P(A|B)P(B)=P(A)P(B).\\ 3.~When~A~and~B~are~mutually~exclusive~then~P(A\cap B)=0~so\\ P(A\cup B)=P(A)+P(B)-P(A\cap B)=P(A)+P(B).\\ When~A~and~B~are~not~mutually~exclusive~then\\~P(A~or~B)=P(A)+P(B)~is~not~true.$