Alice, Bob, and Charlotte are looking for butterflies. they look in three separa

Question

Alice, Bob, and Charlotte are looking for butterflies. they look in three separate parts of a field, so that their probabilities of success do not affect each other . Alice finds 1 butterfly with probability 18%, and otherwise does not find one. Bob finds 1 butterfly with probability 25%, and otherwise does not find one Charlotte finds 1 butterfly with probability 47%, and otherwise does not find one. Let X be the number of butterflies that they find together. Let Y be the number of people who do not find a butterfly. find the joint mass px, rx, y) of X and Y. Use four decimal places in all your probabilities X-0 X-1 X=2 X=3 A Y=0 Y=1 G Y=3

Explanation / Answer

here probability that all of them of them find butterfly =P(y=0)=0.18*0.25*0.47=0.02115

also when all of them catches butterfly P(X=3)=0.02115

also probability that two of them catches butterfly =P(y=1)=P(X=2)

=0.18*0.25*(1-0.47)+0.18*(1-0.25)*0.47+(1-0.18)*0.25*0.47 =0.18365

probability that one of them catches butterfly=P(y=2)=P(X=1)=

0.18*(1-0.25)*(1-0.47)+(1-0.18)*(1-0.25)*0.47+(1-0.18)*0.25*(1-0.47) =0.46925

probability that none of them catches butterfly=P(y=3)=P(X=0)=(1-0.18)*(1-0.25)*(1-0.47)=0.32595

therefore :

x=0 x=1 x=2 x=3 total y=0 0 0 0 0.02115 0.02115 y=1 0 0 0.18365 0 0.18365 y=2 0 0.46925 0 0 0.46925 y=3 0.32595 0 0 0 0.32595 total 0.32595 0.46925 0.18365 0.02115 1

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