Question

In: Statistics and Probability

In your rental shop, you know that on the busiest day you can expect 150 rentals....

In your rental shop, you know that on the busiest day you can expect 150 rentals. You also know that, historically, 60% of your customers rent skis and 40% rent snowboards.

a) If you decide that you only need to have 65 snowboards in stock, what is the probability that you will run out of snowboard rentals on any specific day?

      [ Choose ]            0.2259            0.1794            0.5310            0.0575            0.0664            0.5966            0.0409            0.0000      

b) What if you increase your supply to 70 showboards. Now, what is the probability that you will run out?

      [ Choose ]            0.2259            0.1794            0.5310            0.0575            0.0664            0.5966            0.0409            0.0000      

c) What is the probability that you will rent at most 90 skis on any given day?

      [ Choose ]            0.2259            0.1794            0.5310            0.0575            0.0664            0.5966            0.0409            0.0000      

d) What is the probability that you will rent exactly 90 skis on any given day

Solutions

Expert Solution

    n = 150   …total number of rentals on a day              
   Let X be the number of snowboards rented on a day                  
   p = probability of renting a snowboard                   
        = 0.4   (40%)              
   Then X ~ Binomial(n = 150, p = 0.4)                  
a)   To find P(we run out of snowboards if we stock 65 snowboards)                  
   that is to find P(X > 65)                  
   P(X > 65) = 1 - P(X ≤ 65)                  
   We use Excel function BINOM.DIST to find the probability                  
   P(X > 65) = 1 - BINOM.DIST(65, 150, 0.4, TRUE)                  
       = 1 - 0.8206              
       = 0.1794              
   Answer                  
   0.1794                  
                      
b)   To find P(we run out of snowboards if we stock 70 snowboards)                  
   that is to find P(X > 70)                  
   P(X > 70) = 1 - P(X ≤ 70)                  
   We use Excel function BINOM.DIST to find the probability                  
   P(X > 70) = 1 - BINOM.DIST(70, 150, 0.4, TRUE)                  
       = 1 - 0.9591              
       = 0.0409              
   Answer                  
   0.0409                 
                      
c)   Let Y be the number of skis rented on a given day                  
   p' = probability of renting a ski = 0.6               -60%  
   Then Y ~ Binomial(n = 150, p = 0.6)                  
   To find P(we rent atmost 90 skis on a given day)                  
   that is to find P(Y ≤ 90)                  
   We use Excel function BINOM.DIST to find the probability                  
   P(Y ≤ 90) = BINOM.DIST(90, 150, 0.6, TRUE)                  
       = 0.5310              
   Answer                  
   0.5310                 
                      
d)   To find P(we rent exactly 90 skis on a given day)                  
   that is to find P(Y = 90)                  
   We use Excel function BINOM.DIST to find the probability                  
   P(Y = 90) = BINOM.DIST(90, 150, 0.6, FALSE)                   (For discrete probability, last parameter is FALSE)
       = 0.0664              
   Answer                  
   0.0664                  


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