Question

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. $\frac{240}{729}$
• B. $\frac{192}{729}$
• C. $\frac{256}{729}$
• D. $\frac{496}{729}$

Answer 1

The Correct Option is C

 To solve this problem, we need to determine the probability of getting at least 5 successes in 6 trials given that an experiment succeeds twice as often as it fails.

Step 1: Determine the probability of success and failure

Let the probability of success in a single trial be \( p \) and the probability of failure be \( q \). We are given that the experiment succeeds twice as often as it fails:

\(p = 2q\)

Since the probability of success and failure must add up to 1:

\(p + q = 1\)

Substitute \( p = 2q \) in the equation:

\(2q + q = 1\) \(3q = 1\) \(q = \frac{1}{3}\)

Then, the probability of success is:

\(p = 2q = 2 \times \frac{1}{3} = \frac{2}{3}\)

Step 2: Use the Binomial Probability Formula

The probability of getting exactly \( k \) successes in \( n \) independent Bernoulli trials is given by the binomial probability formula:

\(P(X = k) = \binom{n}{k} p^k q^{n-k}\)

Here, \( n = 6 \), \( p = \frac{2}{3} \), and \( q = \frac{1}{3} \).

We need the probability of at least 5 successes:

\(P(X \geq 5) = P(X = 5) + P(X = 6)\)

Calculate each component:

• For 5 successes:
• For 6 successes:

Step 3: Calculate the probability of at least 5 successes

Now, add the calculated probabilities:

\(P(X \geq 5) = \frac{192}{729} + \frac{64}{729} = \frac{256}{729}\)

Hence, the probability of getting at least 5 successes in 6 trials is \(\frac{256}{729}\).

Conclusion

The correct answer is \(\frac{256}{729}\), which matches the given option.