In the literature it is reported that 10 of acorns of a part

In the literature it is reported that 10% of acorns of a particular oak species have holes due to weevils.   An ecologist is trying to explore whether the probability of sampling acorns with holes fits the binomial distribution with p = .1. She (and her ecology class) take 1000 random samples of acorns across a forested region, with each sample consisting of 5 acorns. They then tally the results and find the following results. They also calculate the binomial proportions that they expected.

      Categories                      Number of samples              Binomial proportions¹

      0 damaged acorn                     640                              .590                            

      1 damaged acorn                     250                              .328

      2 damaged acorns                   50                              .073

      3 damaged acorns                   40                              .008

      4 damaged acorns                   18                              .00045

      5 damaged acorns                      2                              .00001

¹Numbers don’t add up to exactly 1 due to rounding error, but don’t worry about this.

Show calculations of how the number .590 was obtained (in the binomial proportions column).

Solution

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    5      
p = the probability of a success =    0.1      
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.59049 [ANSWER]