Final answer:
The PMF of the number of red M&Ms in a bag can be modeled using a binomial distribution with n=25 and p=1/5. To calculate the probability of no red M&Ms, we use the formula (⁴/₅)^25, as it is a case of P(R=0) in the binomial distribution.
Step-by-step explanation:
The question asks to find the probability mass function (PMF) of the number of red M&Ms (R) in a bag of 25, where each M&M has an equal chance of being red, green, orange, blue, or brown.
The total number of outcomes is 5 (one for each color), and since each piece is equally likely to be of any color, the probability of any M&M being red is ⅕ (1 in 5). The PMF of R can be modeled using the binomial distribution with parameters n=25 (the number of trials or M&Ms in the bag) and p=⅕ (the probability of success, which in this case is drawing a red M&M).
To find the probability of a bag having no red M&Ms, we look at the case where R=0. The formula for the binomial probability is P(R=k) = (n choose k) × p^k × (1-p)^(n-k), where 'k' is the number of successes.
Plugging the values into the formula, we get: P(R=0) = (25 choose 0) × (⅕)^0 × (⁴/₅)^25 = 1 × 1 × (⁴/₅)^25.
The probability that a bag has no red M&Ms can be calculated using this formula. We're trying to calculate P(R=0), which is simply (⁴/₅)^25.