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Mathematics

Probability

Hypergeometric Distribution

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In an election, a voter randomly selects 3 candidates from a list of 4 preferred and 5 non-preferred candidates. What is the probability that the voter selects all preferred candidates?

$P(X = 3) = \frac{{\binom{4}{3} \binom{5}{0}}}{{\binom{9}{3}}}$

In a lot of 20 items with 6 defective ones, what is the probability that a random sample of 5 items contains exactly 2 defective ones?

$P(X = 2) = \frac{{\binom{6}{2} \binom{14}{3}}}{{\binom{20}{5}}}$

In a coding competition, there are 12 advanced level problems and 8 beginner level problems. What is the probability of randomly selecting 5 problems with exactly 3 advanced ones?

$P(X = 3) = \frac{{\binom{12}{3} \binom{8}{2}}}{{\binom{20}{5}}}$

If a classroom has 15 boys and 10 girls, what is the probability that a randomly formed team of 4 students will be all boys?

$P(X = 4) = \frac{{\binom{15}{4} \binom{10}{0}}}{{\binom{25}{4}}}$

Drawing 5 cards without replacement from a standard deck of 52, what is the probability of getting exactly 3 aces?

$P(X = 3) = \frac{{\binom{4}{3} \binom{48}{2}}}{{\binom{52}{5}}}$

From a group of 10 cats and 8 dogs, what is the probability that a randomly chosen group of 6 animals will have exactly 4 cats?

$P(X = 4) = \frac{{\binom{10}{4} \binom{8}{2}}}{{\binom{18}{6}}}$

Given a batch of 20 products with 4 defects, what is the probability that a quality check of 3 products results in finding exactly 1 defective product?

$P(X = 1) = \frac{{\binom{4}{1} \binom{16}{2}}}{{\binom{20}{3}}}$

How likely is it to get 2 sixes in rolling 5 fair dice?

$P(X = 2) = \frac{{\binom{5}{2} \binom{25}{3}}}{{\binom{30}{5}}}$

What is the probability of drawing exactly 2 red balls from an urn containing 7 red and 5 blue balls when drawing 4 balls without replacement?

$P(X = 2) = \frac{{\binom{7}{2} \binom{5}{2}}}{{\binom{12}{4}}}$

At a book club meeting with 7 fiction and 5 non-fiction books, what is the probability that the first 3 books selected are all fiction?

$P(X = 3) = \frac{{\binom{7}{3} \binom{5}{0}}}{{\binom{12}{3}}}$

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