Areport i just made up says that 11% of all flowers sold from a particular flower shop are carnations. if this flower shop sells 316 bouquets on a random day, what is the probability that: a) fewer than 40 are carnations? b) at least 25 are carnations? c) between 25 and 40 are carnations? note: a through c are not answers. they are parts of the question that need to be solved individually.

This is a problem on a binomial probability distribution. A binomial probability distribution has fixed number of trials, n. In this case, n is equal to 316. Also, the trials in a binomial probability distribution must be independent. Each trial must have all outcomes classified into two categories--carnation or not carnation. Lastly, the probability of success remains the same in all trials. 

The binomial probability formula is given by 
     
where 
         n=number of trials
         x=number of successes among n trials
         p=probability of success in any one trial
         q=probability of failure in any one trial (q=1-p)

So, from the given problem, the following quantities are given:
     , 
     
     

Part A
The probability of obtaining fewer than 40 (0 to 39) carnations is the summation of all the binomial probabilities from 0 to 39. That is 
     

Part B
The probability that at least 25 are carnations is the sum of binomial probabilities of x from 25 to 316. That is 
     

Part C
The probability that carnations are between 25 and 40 is given by the sum of individual binomial probabilities from 26 and 39. That is 
      

Therefore, the probability that there are fewer than 40 carnations is 0.8048, at least 25 carnations is 0.9720, and between 25 and 40 is 0.7615.

The scale factor of what ?