Problem:
An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is:

a) 240/729
b) 192/729
c) 256/729
d) 496/729

Solution:
To find the probability of at least 5 successes in 6 trials, we need to consider all the possible outcomes.

Step 1: Determine the Probability of Success and Failure
Let's assume the probability of success in a single trial is 'p' and the probability of failure is 'q'. According to the problem, the experiment succeeds twice as often as it fails. So, we can write:

p = 2q

Step 2: Identify the Possible Outcomes
In 6 trials, the possible outcomes can be represented as combinations of successes (S) and failures (F). Let's denote the number of successes as 'x'. The possible outcomes are:

6 successes: SSSSSS
5 successes: SSSSSF, SSSSFS, SSSSFF, SSSFSS, SSSFSF, SSSFSS, SSSFFS, SSSFFF
4 successes: SSSFF, SSFSS, SSFSF, SSFSS, SSFFS, SSFFF, SFFSS, SFFSF, SFFFS, SFFFF
3 successes: SFFFS, SFFFF, FSSSS, FSSSF, FSSFS, FSSFSS, FSSFFS, FSSFFF, FSFSS, FSFSF, FSFSS, FSFFS, FSFFF
2 successes: FSFFF, FFFFF
1 success: FFFFFF

Step 3: Calculate the Probability of Each Outcome
Now, we can calculate the probability of each outcome using the probabilities of success and failure.

Probability of 'x' successes: p^x * q^(6-x)

Step 4: Calculate the Probability of at Least 5 Successes
To find the probability of at least 5 successes, we need to consider the probabilities of all outcomes with 5 or 6 successes.

Probability of at least 5 successes = Probability of 5 successes + Probability of 6 successes

Probability of 5 successes = (6C5) * (p^5) * (q^(6-5))

Probability of 6 successes = (6C6) * (p^6) * (q^(6-6))

Step 5: Substitute the Values and Calculate
Substitute the values of p and q into the probability equations and calculate the probabilities.

p = 2q
q = 1 - p/3

Probability of 5 successes = (6C5) * ((2q)^5) * (q^(6-5))
Probability of 6 successes = (6C6) * ((2q)^6) * (q^(6-6))

Step 6: Simplify and Add the Probabilities
Simplify the probability equations and add the probabilities to find the final answer.

Probability of at least 5 successes = Probability of 5 successes + Probability of 6 successes

The final answer is option c) 256/729.