Question 952345
Notation/Variable Setup:


G = gets a grant
W = program will work


P(G) = probability she gets the grant
P(G')= probability she doesn't get the grant


P(W|G) = probability program works IF she gets grant
P(W|G') = probability program works IF she does NOT get grant


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"what is the probability of success? " translation: "what is the probability of the program works, ie, what is P(W)? "


To compute P(W), we use the <a href="http://people.reed.edu/~jones/Courses/P02.pdf">law of total probability</a>, which in this specific case means: P(W) = P(W|G)*P(G) + P(W|G')*P(G')


So we need to find the following


P(W|G)
P(W|G')
P(G)
P(G')


So let's pick through the problem to pull out the needed info


"If Sam can get a grant to support her program, she feels that there is a .9 probability that the program will work", so P(W|G) = 0.9


"If she fails to get the grant, she feels that there will only be a .3 probability that it will be successful", so P(W|G') = 0.3


P(G) = 0.6 because "the probability of getting the grant is .6"
P(G') = 0.4 because 1-0.6 = 0.4


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Summary of what we're given


P(W|G) = 0.9
P(W|G') = 0.3
P(G) = 0.6
P(G') = 0.4


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Now we plug all this into the equation P(W) = P(W|G)*P(G) + P(W|G')*P(G') to get...


P(W) = P(W|G)*P(G) + P(W|G')*P(G')


P(W) = 0.9*0.6 + 0.3*0.4


P(W) = 0.54 + 0.12


P(W) = 0.66


The final answer is 0.66

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