Question

In: Statistics and Probability

Your band hires a promoter to promote its shows. A good promoter will result in a...

Your band hires a promoter to promote its shows. A good promoter will result in a sold-out show 90% of the time. A promoter who isn’t good will result in a sold-out show 55% of the time. You initially figure that there is a 70% chance that this promoter is good. Your first show sells out, then the next shows fails to sell out. What is the new probability that the promoter is good? Round your answer to the nearest 1%.

Solutions

Expert Solution

P[ promoter is good ] = 70% = 0.7

P[ promoter is not good ] = 1 - P[ promoter is good ] = 1 - 0.7 = 0.3

P[ sold out | promoter is good ] = 90% = 0.9

P[ fail to sell out | promoter is good ] = 1 - P[ sold out | promoter is good ] = 1 - 0.9

P[ sold out | promoter is not good ] = 55% = 0.55

P[ fail to sell out | promoter is not good ] = 1 - P[ sold out | promoter is not good ] = 1 - 0.55 = 0.45

P[ first show sells out, then the next shows fails to sell out | promoter is good ] = P[ sold out | promoter is good ]*P[ fail to sell out | promoter is good ]

P[ first show sells out, then the next shows fails to sell out | promoter is good ] = 0.9*0.1 = 0.09

P[ first show sells out, then the next shows fails to sell out | promoter is not good ] = P[ sold out | promoter is not good ]*P[ fail to sell out | promoter is not good ]

P[ first show sells out, then the next shows fails to sell out | promoter is not good ] = 0.45*0.55

P[ first show sells out, then the next shows fails to sell out | promoter is not good ] = 0.2475

P[ first show sells out, then the next shows fails to sell out ] = P[ first show sells out, then the next shows fails to sell out | promoter is good ]*P[ promoter is good ] + P[ first show sells out, then the next shows fails to sell out | promoter is not good ]*P[ promoter is not good ]

P[ first show sells out, then the next shows fails to sell out ] = 0.09*0.7 + 0.2475*0.3

P[ first show sells out, then the next shows fails to sell out ] = 0.063 + 0.07425

P[ first show sells out, then the next shows fails to sell out ] = 0.13725

P[ promoter is good | first show sells out, then the next shows fails to sell out ] = P[ first show sells out, then the next shows fails to sell out | promoter is good ]*P[ promoter is good ] / P[ first show sells out, then the next shows fails to sell out ]

P[ promoter is good | first show sells out, then the next shows fails to sell out ] = 0.09*0.7 / 0.13725

P[ promoter is good | first show sells out, then the next shows fails to sell out ] = 0.063/0.13725

P[ promoter is good | first show sells out, then the next shows fails to sell out ] = 0.459

P[ promoter is good | first show sells out, then the next shows fails to sell out ] = 45.9%

orchestra answered 1 year ago
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