Final answer:
To find the probability of at least 5 successes in 6 trials, where success is twice as likely as failure, we use the binomial probability formula for exactly 5 and 6 successful outcomes, summing the results.
Step-by-step explanation:
The question is about calculating the probability of an experiment succeeding in a given number of trials. Specifically, we want to find the probability that there will be at least 5 successes in 6 trials, given that the experiment succeeds twice as often as it fails.
We first need to determine the probability of success (p) and failure (q). Since the experiment succeeds twice as often as it fails, let's assume that the probability of failure is 'x'.
Thus, the probability of success is '2x'. Since these are the only two outcomes and their probabilities must sum to 1, we have x + 2x = 1, which gives us x = 1/3 and 2x = 2/3. Now, we can use the binomial probability formula to calculate the probability of exactly 5 successes (P(X=5)) and exactly 6 successes (P(X=6)) in 6 trials:
P(X=k) = C(n, k) * p^k * q^(n-k)
where C(n, k) is the number of combinations of n items taken k at a time.
The probability of at least 5 successes is the sum of the probabilities of exactly 5 and exactly 6 successes:
P(X ≥ 5) = P(X=5) + P(X=6)
We can calculate:
C(6, 5) = 6
C(6, 6) = 1
Hence,
P(X=5) = C(6, 5) * (2/3)^5 * (1/3)^1
P(X=6) = C(6, 6) * (2/3)^6 * (1/3)^0
After calculating both probabilities, we sum them to find the final probability of at least 5 successes.