Final answer:
The student's question regards calculating probabilities for a tennis player within a binomial distribution. One calculation is for the probability of winning exactly 4 out of 6 matches, and another is for winning fewer than 3 matches. Both probabilities are calculated using the binomial probability formula.
Step-by-step explanation:
This is a binomial distribution problem because it meets the criteria for a binomial experiment: only two possible outcomes (win or lose), a constant probability of success in each trial (p = 0.70), and each match is an independent event. To calculate the probability that Annie wins exactly 4 of the 6 tennis matches, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * q^(n-k)
where:
- C(n, k) is the number of combinations (n choose k)
- p is the probability of winning a single match
- q is the probability of losing a single match (q = 1 - p)
- n is the total number of matches played
- k is the number of matches won
In this case, n = 6, p = 0.70, q = 0.30, and we want to find P(X = 4).
For the second part, to find the probability that Annie wins fewer than 3 matches:
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
By calculating these individual probabilities using the binomial formula, you can accumulate the total probability for winning fewer than 3 matches.