Probability with Combinatorics - Flashcard set - Combinatorics

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Given 15 billiard balls, what is the probability that randomly picking 5 balls includes the 8 ball?

$\frac{{14 \choose 4}}{{15 \choose 5}} = \frac{1001}{3003} \approx 0.3333333333$

With a standard deck of cards, what is the probability that a random 5-card hand has exactly one pair (and no three-of-a-kind, four-of-a-kind, or two pairs)?

$\frac{{13 \choose 1} \times {4 \choose 2} \times {12 \choose 3} \times 4^3}{{52 \choose 5}} = \frac{1098240}{2598960} \approx 0.4225690276$

How likely is it to draw 2 aces from a standard deck of 52 cards in succession without replacement?

$\frac{4}{52} \times \frac{3}{51} = \frac{1}{221}$

What is the probability of 5 fair coin flips resulting in exactly 3 heads?

$\frac{{5 \choose 3} \times 0.5^3 \times 0.5^2}{{1}} = \frac{10}{32}$

What is the probability of being dealt a straight (excluding straight flush and royal flush) in a 5-card poker hand?

$\frac{{10 \choose 1} {4^5 - 4}{{52 \choose 5}} = \frac{10200}{2598960} \approx 0.00392465$

If you shuffle a deck of cards, what is the probability that the top four cards are two black cards followed by two red cards?

$\frac{{26 \choose 2} \times {26 \choose 2} \times 2!}{{52 \choose 4}} = \frac{135850}{270725} \approx 0.501813736$

Three cards are drawn successively with replacement from a deck of 52 cards. What is the probability of drawing 3 kings in a row?

$\left(\frac{4}{52}\right)^3 = \frac{1}{21952}$

In a lottery where 6 numbers are chosen from 49, what is the probability of picking all 6 correctly?

$\frac{{1}}{{49 \choose 6}} = \frac{1}{13983816}$

What is the probability of rolling a total of 7 with two fair six-sided dice?

$\frac{{6}}{{36}} = \frac{1}{6}$

In a class of 30 students, 10 are to be selected at random for a field trip. What is the probability that two specific students will both be selected?

$\frac{{28 \choose 8}}{{30 \choose 10}} = \frac{3003}{43560} \approx 0.068959731$

From a standard deck, what is the probability of drawing 3 cards, all of different ranks?

$\frac{{13 \choose 3} \times 4^3}{{52 \choose 3}} = \frac{13244}{22100} \approx 0.59891429$

If 4 people are to be selected from a group of 8 to form a committee, what is the probability that two specific persons will be included in the committee?

$\frac{{6 \choose 2}}{{8 \choose 4}} = \frac{15}{70} = \frac{3}{14}$

A group of 10 friends is to be seated randomly around a circular table. What is the probability that two particular friends will sit next to each other?

$\frac{{2! \times 8!}}{{9!}} = \frac{2}{9}$

What is the probability that a 5-card poker hand contains cards from only one suit?

$\frac{{4 \choose 1} \times {13 \choose 5}}{{52 \choose 5}} = \frac{5148}{2598960} \approx 0.0019807923$

In a raffle with 100 tickets, where 5 distinct prizes will be given out, what is the probability that a person who buys 10 tickets will win exactly 2 prizes?

$\frac{{10 \choose 2} \times {90 \choose 3}}{{100 \choose 5}} = \frac{4050}{752875} \approx 0.0053805765$

When rolling 3 six-sided dice, what is the probability of rolling a sum of 10?

$\frac{{27}}{{216}} = \frac{1}{8}$

What is the probability of drawing a full house (3 cards of one rank and 2 cards of another rank) in a 5-card poker hand?

$\frac{{13 \choose 1} {4 \choose 3} {12 \choose 1} {4 \choose 2}}{{52 \choose 5}} = \frac{3744}{2598960} \approx 0.00144058$

What is the probability that a 4-digit PIN code, with digits from 0 to 9, has no repeated digits?

$\frac{{10 \choose 1} \times {9 \choose 1} \times {8 \choose 1} \times {7 \choose 1}}{10^4} = \frac{5040}{10000} = 0.504$

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