Final answer:
To find the probability that Laura wins exactly three out of the next four matches against Jennifer, we can use the binomial probability formula.
Step-by-step explanation:
To find the probability that Laura wins exactly three out of the next four matches against Jennifer, we can use the binomial probability formula. The formula is:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of getting k successes in n trials
- C(n, k) is the number of combinations of n items taken k at a time
- p is the probability of a success in a single trial
- n is the total number of trials
- k is the number of successes
In this case, Laura winning a match is a success and Jennifer winning a match is a failure. We want to find the probability of exactly 3 successes out of 4 trials, so n = 4 and k = 3. The probability of Laura winning a match is p = 2/3.
Therefore, plugging these values into the formula, we have:
P(X=3) = C(4, 3) * (2/3)^3 * (1-(2/3))^(4-3)
Simplifying this equation gives us:
P(X=3) = 4 * (2/3)^3 * (1/3)^1
P(X=3) = 4 * 8/27 * 1/3
P(X=3) = 32/81
So the probability that Laura wins exactly three of the next four matches she plays against Jennifer is 32/81.