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Probability

Question
CBSEENMA12035719

There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tails 40% of the times. One of The three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?

Solution
Let space straight E subscript 1 comma space straight E subscript 2 comma space straight E subscript 3 space and space straight A space be space the space events space defined space as space follows colon
straight E subscript 1 equals space space Choosing space 2 space headed space coin
straight E subscript 2 equals Choosing space coin space with space 75 percent sign space chance space of space getting space heads
straight E subscript 3 equals Choosing space coin space with space 40 percent sign space chance space of space getting space heads
straight A space equals space Getting space heads
Then space straight P left parenthesis straight E subscript 1 right parenthesis space equals space straight P left parenthesis straight E subscript 2 right parenthesis space equals space straight P left parenthesis straight E subscript 3 right parenthesis space equals space 1 third
Also comma space straight P left parenthesis straight A divided by straight E subscript 1 right parenthesis space equals space 1 comma space space straight P left parenthesis straight A divided by straight E subscript 2 right parenthesis space equals space 75 over 100 equals 3 over 4 comma space space straight P left parenthesis straight A divided by straight E subscript 3 right parenthesis space equals space 40 over 100 equals 2 over 5
Required space Probability
equals straight P left parenthesis straight E subscript 1 divided by straight A right parenthesis
equals space fraction numerator straight P left parenthesis straight E subscript 1 right parenthesis thin space straight P left parenthesis straight A divided by straight E subscript 1 right parenthesis right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis space straight P left parenthesis straight A divided by straight E subscript 1 right parenthesis plus straight P left parenthesis straight E subscript 2 right parenthesis thin space straight P left parenthesis straight A divided by straight E subscript 2 right parenthesis plus straight P left parenthesis straight E subscript 3 right parenthesis space straight P left parenthesis straight A divided by straight E subscript 3 right parenthesis end fraction
equals fraction numerator begin display style 1 third end style cross times 1 over denominator begin display style 1 third end style cross times 1 plus begin display style 1 third end style cross times begin display style 3 over 4 end style plus begin display style 1 third end style cross times begin display style 2 over 5 end style end fraction
equals space fraction numerator begin display style 1 third end style over denominator begin display style 1 third end style plus begin display style 1 fourth end style plus begin display style 2 over 15 end style end fraction
equals fraction numerator begin display style 1 third end style over denominator begin display style 43 over 60 end style end fraction space equals 1 over 860

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