Question 1021049: the probability that an executive is promoted to a higher position is 5/8. in case he is promoted, he will go on vacation with the probability of 5/6 . however if he is not promoted there is a chance of 1/3 that the executive will go on a vacation. 1. find the probability that the executive will go on vacation. 2. given that he does go on a vacation, find the probability that he has been promoted
Answer by mathmate(429)   (Show Source):
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Question:
the probability that an executive is promoted to a higher position is 5/8. in case he is promoted, he will go on vacation with the probability of 5/6 . however if he is not promoted there is a chance of 1/3 that the executive will go on a vacation. 1. find the probability that the executive will go on vacation. 2. given that he does go on a vacation, find the probability that he has been promoted
Solution:
P=promotion
V=vacation
First make a tree diagram
(5/8) promotion - P(V)=5/6
/
\
(3/8)no promotion-P(V)=1/3
or, assuming it is a two step experiment,
P(PV)=(5/8)(5/6)=25/48
P(P~V)=(5/8)(1/6)=5/48
P(~PV)=(3/8)(1/3)=1/8=6/48
P(~P~V)=(3/8)(2/3)=1/4=12/48
Check(25+5+6+12)/48=1 ok.
1.
P(V)
=P((P∪~P)∩V) [law of total probability]
=P(PV)+P(~PV)
=25/48+6/48=31/48
2.
P(P|V)
=P(P∩V)/P(V) [by definition of conditional probability]
=P(PV)/P(V)
=(25/48)/(31/48)
=25/31
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