Final answer:
The question deals with calculating probabilities using the binomial distribution in Mathematics. No exact probabilities can be given without further information, but the methods to find these probabilities (at least, exactly, and less than a certain number) relate to how we sum the probabilities for binomial outcomes.
Step-by-step explanation:
The subject of this question is Mathematics, specifically probability. Given that 4 out of every 7 dentists recommend a particular toothpaste, the probability that a randomly chosen dentist recommends it is 4/7. When six dentists are chosen, we can use the binomial probability formula to calculate the probabilities for each of the scenarios provided:
- At least three dentists: To find the probability that at least three out of six dentists recommend the toothpaste, we must calculate the probability for three, four, five, and six dentists recommending it, and then sum those probabilities together.
- Exactly three dentists: The probability that exactly three dentists recommend the toothpaste is found by using the exact count in the binomial formula.
- Less than three dentists: This is the probability of either zero, one, or two dentists recommending the toothpaste. We would calculate each probability and sum them to get the total probability for 'less than three'.
However, without a specific probability distribution or additional details, we cannot provide the exact numerical probabilities, and therefore no conclusion can be drawn.