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Which of the following can be the valid assignment of probability for outcomes of sample space, `S = {W_(1), W_(2), W_(3)}, {" where " W_(1), W_(2) " and " W_(3)` are mutually exclusive events
Assignment
`{:(,w_(1),w_(2),w_(3)),((a),0.4,0.2,0.8),((b),0,0,1),((c),-1/2,3/4,1/2),((d),1/2,1/2,1/4):}`

Text Solution

Verified by Experts

(a) We have, `P(W_(1)) = 0.4, P(W_(2)) = 0.2 " and " P(W_(3)) = 0.8`
`therefore 0leP(W_(i))le1, " for each " W_(i) in S`
But, `P(W_(1)) + P(W_(2)) +P(W_(3)) = 0.4 + 0.2 + 0.8 = 1.4 gt 1`
Hence, this assignment is not valid.
(b) `P(W_(1)) = 0, P(W_(2)) = 0, P(W_(3)) = 1`
Here, `0leP(W(i))le1, " where " W_(i) in S " and " P(W_(1))+P(W_(2))+P(W_(3))=1`
Hence this assignment is valid.
(c) Here, `P(W_(1))lt0`
So, axiom 1 is not satisfied
Hence, this assignment is not valid.
(d) Again, `P(W_(1))+P(W_(2))+P(W_(3)) = (1)/(2)+(1)/(2)+(1)/(4)=(5)/(4)gt0`
Hence, this assignment is not valid.
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