Final answer:
The question is about calculating binomial probabilities where the number of dentists recommending a specific toothpaste is considered. The probability of individual recommendations is 4/9, and calculations are made for different numbers of dentists recommending the product out of a sample of 8.
Step-by-step explanation:
The subject of this question is Mathematics, specifically in the area of probability. We are dealing with the probability of a certain number of dentists recommending a toothpaste given a fixed probability for individual recommendations.
To find the probability that exactly five dentists recommend Mintyfresh Toothpaste, we use the binomial probability formula since each recommendation is an independent event:
- P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
- For 'Less than seven dentists recommend Mintyfresh', we need to calculate the cumulative probability for 0 to 6 recommendations.
- For 'At least six dentists recommend Mintyfresh', we would look at the complements for 0 to 5 recommendations to add up the probabilities for 6, 7, and 8 recommendations.
In this scenario, 'p' represents the probability of individual recommendation, which is 4/9, 'n' is the total number of dentists selected (8 dentists), and 'k' would be the number of dentists we are interested in, which varies for each part of the question.