Probability: 'At Least' Questions

What is the minimum number of people that need to be chosen at random in order to have a probability of 50% or more that at least one of them is born in a leap year?

Incorrect. [[snippet]] The probability that a randomly selected person was born in a leap year is $$\frac{1}{4}$$. Therefore, the probability that that person was _not_ born in a leap year is $$1 - \frac{1}{4} = \frac{3}{4}$$.

Correct. [[snippet]] The probability that a randomly selected person was born in a leap year is $$\frac{1}{4}$$. Therefore, the probability that that person was _not_ born in a leap year is $$1 - \frac{1}{4} = \frac{3}{4}$$. If you pick 2 people, the probability that neither of them was born in a leap year is $$\frac{3}{4}\times \frac{3}{4} = \frac{9}{16}$$. Therefore, the probability that at least one of them was born in a leap year is $$1- \frac{9}{16} = \frac{7}{16}$$. Since $$0.5 = \frac{8}{16}$$, this is _less_ than 50%. If you pick 3 people, the probability that none of them was born in a leap year is $$\frac{3}{4} \times \frac{3}{4}\times \frac{3}{4} = \frac{27}{64}$$. Therefore, the probability that at least one of them was born in a leap year is $$1 - \frac{27}{64} = \frac{37}{64}$$. Since $$0.5 = \frac{32}{64}$$, this is _more_ than 50%.

Incorrect. [[snippet]] The probability that a randomly selected person was born in a leap year is $$\frac{1}{4}$$. Therefore, the probability that that person was _not_ born in a leap year is $$1 - \frac{1}{4} = \frac{3}{4}$$. If you pick 2 people, the probability that neither of them was born in a leap year is $$\frac{3}{4}\times \frac{3}{4} = \frac{9}{16}$$. Therefore, the probability that at least one of them was born in a leap year is $$1- \frac{9}{16} = \frac{7}{16}$$. Since $$0.5 = \frac{8}{16}$$, this is _less_ than 50%.

Incorrect. [[snippet]]

Incorrect. [[snippet]]

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