Jim has a barber shop. On a given day, if 3customers show upin total, after givi

Question

Jim has a barber shop. On a given day, if 3customers show upin total, after giving a hair to the 3rd customer Jim closes theshop and goes home early. So Jim cannot have more than 3 customersin a day. Suppoese that having 3customers is twice as likely ashaving 2 customers, which is 3 times as likely as hav ing 1 cumter,which is four times as likely as having no customers. a) what is the probability that Jim goes home early? =>24/41 b) given that at least 1 customer showed up, what is theprobability that Jim goes home early? ==> 24/40 c) given that Jim goes home early, what is the probabilitythat no customers showed up? ==> 0 Jim has a barber shop. On a given day, if 3customers show upin total, after giving a hair to the 3rd customer Jim closes theshop and goes home early. So Jim cannot have more than 3 customersin a day. Suppoese that having 3customers is twice as likely ashaving 2 customers, which is 3 times as likely as hav ing 1 cumter,which is four times as likely as having no customers. a) what is the probability that Jim goes home early? =>24/41 b) given that at least 1 customer showed up, what is theprobability that Jim goes home early? ==> 24/40 c) given that Jim goes home early, what is the probabilitythat no customers showed up? ==> 0

Explanation / Answer

Let X=number of customers   X=0,1,2,3 Given P(3)=2P(2); P(2)=3P(1);P(1)=4P(0) In terms of P(0), we have P(3)=2(3)(4)P(0)=24P(0); P(2)=12P(0); P(1)=4P(0) Now the P's must sum to 1, so P(0)+P(1)+P(2)+P(3)=1 P(0)=1/41 a)Hence he goes home early if X=3 P(3)=24/41 c)Given at least one customer shows up we have 1-P(0)=40/41,hence P(X=3/at least 1 showup)=P(X=3)/(1-P(0))=(24/41)/(40/41)=24/40 c)0

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