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Aishwarya studies either computer science or mathematics everyday. If she studies computer science on a day, then the probability that she studies mathematics the next day is $0.6$. If she studies mathematics on a day, then the probability that she studies computer science the next day is $0.4$. Given that Aishwarya studies computer science on Monday, what is the probability that she studies computer science on Wednesday?

  1. $0.24$
  2. $0.36$
  3. $0.4$
  4. $0.6$
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I'll use obvious notations.

$P(M|C)=0.6$

$P(C|M)=0.4$

 

Since she studies either of $M$ or $C$ everyday $P(C|C)=0.4$

 

If she studies $C$ on Monday, we want her to study $C$ on Wednesday, too.

We can do this via:

$C \rightarrow C \rightarrow C$ or $C \rightarrow M \rightarrow C$

$0.4*0.4+0.6*0.4$

=> $0.16+0.24$

=> $0.4$
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Monday                     Tuesday            Wednesday

 

                                                            CS(0.4)    Let cs1=Studying computer science on Monday,cs2=studying computer science on  Tuesday

                                 CS(0.4)                                 cs3=studying computer science on Wednesday,math1=studying math on Monday                          

                                                            Math(0.6)   math2=studying math on Tuesday,math3=studying math on Wednesday

 CS                                                                        Now P(studying computer science on Wednesday given that studying cs on Monday)

                                                            CS(0.4)            

                                 Math(0.6)                              ={P(cs1∩cs2∩cs3)+P(cs1∩math2∩cs3)}/P(cs1)

                                                            Math(0.6)   ={P(cs3|cs1∩cs2)*P(cs1∩cs2)+P(cs3|cs1∩math2)*P(cs1∩math2)}/P(cs1)

                                                                                ={P(cs3|cs1∩cs2)*P(cs2|cs1)*P(cs1)+P(cs3|cs1∩math2)*P(math2|cs1)P(cs1)}/P(cs1)

                                                                                 = P(cs3|cs1∩cs2)*P(cs2|cs1)+P(cs3|cs1∩math2)*P(math2|cs1)

                                                            CS(0.4)        =0.4*0.4+0.4*0.6

                                 CS(0.4)                                   =0.4 (C)

                                                            Math(0.6)

 Math   

                                                            CS(0.4)

                                Math(0.6)

                                                            Math(0.6)
Answer:

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