Question

Imagine an archer is able to hit the bull’s-eye 82% of the time. Assume each shot is independent of all others. If the archer shoots 10 arrows, then the probability that every arrow misses the bull’s-eye is a value between: ?

In problem above: The expected value of the number of arrows that hit the bull’s-eye, is exactly equal to: ?

In problem above: The probability that the archer hits the bull’s-eye more often than they miss, is a value between: ?

Please show all steps and how to calculate.

Answer 1

    n = 10   number of arrows the archer shoots                  
   p = 0.82   82% hit the bull's eye                  
   Let X be the number of arrows that hit the bull's eye                      
   X ~ Binomial distribution with n = 10 and p = 0.82                      
                          
a)     To find P(every arrow misses the bull's eye)                      
   that is to find P(X=0)                      
   We use the Excel function BINOM.DIST to find the probability                      
   P(X=0) = BINOM.DIST(0,10,0.82,FALSE)                      
                    = 0.000000038                     
   P(every arrow misses the bull's eye) is between 0 and 0.000001                     
                          
b)     The expected value of a binomial distribution is                       
   E(X) = np                      
            = 10 * 0.82                      
            = 8.2                      
   Expected value of number of arrows that hit the bull's eye = 8.2    hits                 
                          
c)     To find P(archer hits bull's eye more often than misses)                      
   that is to find P(archer hits bull's eye more than 50% of the time)                      
   that is to find P(archer hits bull's eye more than 5 out of 10 times)                      
   that is to find P(X > 5)                      
   P(X > 5) = 1 - P(X <= 5)                      
   We use the Excel function BINOM.DIST to find the probability                      
   P(X > 5) = 1 - BINOM.DIST(5,10,0.82,TRUE)                   (for cumulative probability last parameter is TRUE)  
                    = 1 - 0.0213                      
                    = 0.9787                      
   P(archer hits bull's eye more often than misses) = 0.9787