A die is tossed 6 times and getting “1 or 5” is considered a success. The probability of getting at least one success in six tosses is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 665/729
On one toss, a success is 1 or 5:
\( \displaystyle p = \frac{2}{6} = \frac{1}{3},\quad q = 1 - p = \frac{2}{3} \)
The number of successes X in 6 tosses is binomial with n = 6. “At least one” is the complement of “none”:
\(P(X \ge 1) = 1 - P(X = 0) = 1 - q^6\)
\( \displaystyle = 1 - \left(\frac{2}{3}\right)^6 = 1 - \frac{64}{729} \)
\( \displaystyle = \frac{665}{729} \)
Why the other options are incorrect
Option (1) 1/3 — Incorrect. This is the probability of a success on a single toss.
Option (2) 1/729 — Incorrect. This is \(\left(\frac{1}{3}\right)^6\), the probability of a success on all six tosses.
Option (3) 728/729 — Incorrect. This is \(1 - \left(\frac{1}{3}\right)^6\), which uses p in place of q: it is the probability of at least one failure.