To solve this problem, we will use the Binomial probability formula. Binomial probability is used when there are exactly two mutually exclusive outcomes of a trial. These outcomes are labelled "success" and "failure". We are trying to find out the probability of an exact number of "successes" from a fixed number of trials.
Step 1:
Our Random Variable (X) is defined as 'the number of brown M&M's in a bag'.
Step 2:
Considering the proportion of brown M&M's (13% or 0.13 when considering it as a proportion) and the total number of M&M's in a bag (69), we want to find the exact probability of having no brown M&M's in a bag.
Step 3:
For this kind of problems, we use the Binomial probability formula:
P(X=k) = C(n, k) * (p^k) * (1-p)^(n-k)
where,
n = total number of trials
k = number of successes we are interested in
p = probability of success in a single trial
C(n, k) = Binomial coefficient, ways of choosing k successes from n trials
Given:
n = 69 (total number of M&M's in a bag)
k = 0 (number of brown M&M's, which we are interested in)
p = 0.13 (proportion of brown M&M's)
Step 4:
Plugging numbers into the formula:
P(X=0) = C(69, 0) * (0.13^0) * (1-0.13)^(69-0)
Step 5:
Solving the above calculation, we obtain the probability as 0.0001 (rounded to 4 decimal places).
So, the probability of having no brown M&M's in a bag of 69 is approximately 0.0001. This means it's very rare to find a bag of 69 M&M's without a single brown one, considering the proportion of brown M&M's is 13%.