The fine print on an instant lottery ticket claims that one in nine tickets win a prize. What is the probability that you win at least twice if you purchase ten tickets?
2026-03-28 15:55:40.1774713340
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Probability to win
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The probability that you win at least twice is equal to one minus the probability that you win 0 or 1 time. In notation: $$P(\text{win at least } 2) = 1 - P(\text{win } 1 \text{ or } 0)$$
The probability that you win $0$ times is $$P(0) = \binom{10}{0}\left(\frac{1}{9}\right)^{10}$$
Now all that's left to do is find the probability that you win $1$ time. Can you take it from here?
The value you have, $\frac{1}{9}$, is a probability of success, $p$. Denote the full set of outcomes $\Omega$. The set of outcomes you are interested in is $\Omega_s$. Now try to find $P(\Omega) - P(\Omega_s^c)$ where $\Omega_s^c$ is the set of outcomes complimentary to the one you are interested in using Binomial distribution.