Question

Suppose you have a 1/10 chance of winning with a scratch-off lottery ticket. If you buy 4 tickets, what is the probability of winning with all 4?

Answer 1

The proper way to answer such questions is to recognize the probability distribution first. Although you could simply take the probability and raise it to the fourth power, this idea will not work for most questions.

The scenario given fulfills the requirements as a binomial distribution. There is a probability of failure and a probability of success, where the probabilities are constant for each defined trial.
\(\text{B(n,p,k)}=\binom{n}{k}p^k q^{n-k} \)

\(p=\frac{1}{10}\)
\(∴q=1-p=1-\frac{1}{10}=\frac{9}{10}\)

\(\text{B(4,0.1,4)}=\binom{4}{4}(\frac{1}{10})^4 (\frac{9}{10})^{4-4}=(\frac{1}{10})^4 \)