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Two dice are thrown. Event #A# is "the sum of two dice is #7#" and Event #B# is "at least one die is #6#". Are #A# and #B# independent?

Answer 1

Oliver Aldrich

See the Discussion in the Explanation.

Clearly, we have, as the Sample Space S, associated with the

given Random Experiment,

#S={1,2,...,6}xx{1,2,...,6}., and, n(S)=36.#

#"The Event A="{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}.#

#:. n(A)=6 rArr P(A)=(n(A))/(n(S))=6/36=1/6.......................(1).#

#"The Event B="{(1,6),(2,6),(3,6),(4,6),(5,6),(6,6),(6,1),(6,2),(6,3),(6,4),(6,5)}.#

#:. n(B)=11 rArr P(B)=11/36.............................................(2).#

Note that, #AnnB={(1,6),(6,1)}.#

#:. P(AnnB)=2/36=1/18..................................................(3).#

#(1),(2),(3) rArr P(AnnB)=1/8ne1/6*11/36=11/216=P(A)P(B).#

This shows that the Events #A, and, B# are not Independent.

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