Distributions, Parameters, Probabilities - Extra Practice

Question 1

Sir Fluffington is a deeply indecisive cat. His owners have lined up 5 identical cardboard boxes, labeled 1 through 5, and every afternoon he picks one completely at random to nap in. \(X\) is the number on the box he picks today.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Discrete uniform. Every box is equally likely to be picked, and the boxes are labeled 1, 2, 3, and so on.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim\) Discrete Uniform\((5)\), so \(n = 5\).

Question 2

A bag of Scrabble tiles holds 100 tiles, 12 of which are E’s. You draw 7 tiles to start the game. \(X\) is the number of E’s you draw.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Hypergeometric. Tiles you draw don’t go back in the bag, so you’re drawing without replacement, and you count how many are E’s.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim HG(100, 12, 7)\), so \(N = 100\), \(G = 12\) and \(n = 7\).

3. Write an expression for the probability that you draw exactly 2 E’s.

Show answer

\[P(X = 2) = \dfrac{\binom{12}{2}\binom{88}{5}}{\binom{100}{7}}\]

Here, \(N\) is the total number of tiles (100), \(G\) is the number of E’s (the thing we’re counting - so 12), and \(n\) is the number of tiles you draw (7).

\[P(X = k) = \dfrac{\binom{G}{k}\binom{N-G}{n-k}}{\binom{N}{n}}\]

Question 3

A carnival prize wheel has 24 equal slots, numbered 1 through 24. You give it one big spin. \(X\) is the number it stops on.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Discrete uniform. The slots are equal, so every number on the wheel is equally likely, and they’re numbered 1, 2, 3, and so on.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim\) Discrete Uniform\((24)\), so \(n = 24\).

Question 4

A fearsome dragon guards a hoard of 40 golden eggs. 30 are pure gold, and 10 are identical-looking chocolate eggs wrapped in gold foil (given to the dragon by his sweetheart). One night, a brave knight sneaks into the cave as the dragon sleeps, quickly swipes 6 random eggs, and runs home gleefully to examine his haul. Let \(X\) be is the number of chocolate eggs the knight found in his loot.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Hypergeometric. The knight takes eggs without putting any back, and you count how many of them are chocolate.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim HG(40, 10, 6)\), so \(N = 40\), \(G = 10\) and \(n = 6\).

3. Write an expression for the probability that the knight escapes with at most 1 chocolate egg.

Show answer

\[P(X \le 1) = P(X = 0) + P(X = 1) = \dfrac{\binom{10}{0}\binom{30}{6}}{\binom{40}{6}} + \dfrac{\binom{10}{1}\binom{30}{5}}{\binom{40}{6}}\]

Here, \(N\) is the total number of eggs (40), \(G\) is the number of chocolate eggs (the thing we’re counting - so 10), and \(n\) is the number of eggs the knight stole (6).

\[P(X = k) = \dfrac{\binom{G}{k}\binom{N-G}{n-k}}{\binom{N}{n}}\]

Question 5

A carnival duck pond has 50 rubber ducks floating in it, and 10 of them have “WINNER” written on the bottom. You pick up one duck. \(X = 1\) if it says “WINNER” and \(X = 0\) if not.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Bernoulli. There’s a single pick with two outcomes, and \(X\) is 1 for a success (a winning duck) and 0 otherwise.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim Bernoulli(0.2)\), so \(p = 10/50 = 0.2\).

Question 6

Each time you play a claw machine, it grabs a prize with probability 1/8, independently of other plays. You play 16 times. \(X\) is the number of prizes you win.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Binomial. The plays are independent, each has the same chance of grabbing a prize, and \(X\) counts the prizes over a fixed number of plays.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim Bin(16, 1/8)\), so \(n = 16\) and \(p = 1/8\).

3. Write an expression for the probability that you win at least one prize.

Show answer

\[P(X \ge 1) = 1 - P(X = 0) = 1 - \left(\dfrac{7}{8}\right)^{16}\]

Here, \(n\) is the number of plays (16) and \(p\) is the chance that each play grabs a prize (1/8).

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]

Question 7

You’re dealt 5 cards from a well-shuffled standard 52-card deck. \(X\) is the number of hearts in your hand.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Hypergeometric. Cards are dealt without replacement, and you count how many of them are hearts.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim HG(52, 13, 5)\), so \(N = 52\), \(G = 13\) and \(n = 5\).

3. Write an expression for the probability that your hand has at least 4 hearts.

Show answer

\[P(X \ge 4) = P(X = 4) + P(X = 5) = \dfrac{\binom{13}{4}\binom{39}{1}}{\binom{52}{5}} + \dfrac{\binom{13}{5}\binom{39}{0}}{\binom{52}{5}}\]

Here, \(N\) is the total number of cards in the deck (52), \(G\) is the number of hearts (the thing we’re counting - so 13), and \(n\) is the number of cards you’re dealt (5).

\[P(X = k) = \dfrac{\binom{G}{k}\binom{N-G}{n-k}}{\binom{N}{n}}\]

Question 8

At Billy’s Goat Yoga, 9 goats wander around the room during class. Each goat, independently of the others, has a 30% chance of climbing onto someone’s back during the “downward dog” pose. \(X\) is the number of goats that climb onto a back during that pose.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Binomial. Each goat is an independent trial with the same chance of climbing, and \(X\) counts the climbers among a fixed number of goats.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim Bin(9, 0.3)\), so \(n = 9\) and \(p = 0.3\).

3. Write an expression for the probability that exactly 3 goats climb onto a back.

Show answer

\[P(X = 3) = \binom{9}{3}(0.3)^3(0.7)^6\]

Here, \(n\) is the number of goats (9) and \(p\) is the chance that each goat climbs onto a back (0.3).

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]

Question 9

You roll a fair six-sided die 10 times. \(X\) is the number of sixes.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Binomial. Each roll is an independent trial with the same chance of a six, and \(X\) counts the sixes over a fixed number of rolls.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim Bin(10, 1/6)\), so \(n = 10\) and \(p = 1/6\).

3. Write an expression for the probability that you don’t roll a single six.

Show answer

\[P(X = 0) = \left(\dfrac{5}{6}\right)^{10}\]

Here, \(n\) is the number of rolls (10) and \(p\) is the chance that each roll is a six (1/6).

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]

Question 10

A drawer holds 14 loose socks, 4 of them striped. In the dark, you pull out 3 socks. \(X\) is the number of striped socks you pull out.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Hypergeometric. The socks you pull out don’t go back in the drawer, and you count how many are striped.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim HG(14, 4, 3)\), so \(N = 14\), \(G = 4\) and \(n = 3\).

3. Write an expression for the probability that you pull out at least one striped sock.

Show answer

\[P(X \ge 1) = 1 - P(X = 0) = 1 - \dfrac{\binom{4}{0}\binom{10}{3}}{\binom{14}{3}}\]

Here, \(N\) is the total number of socks (14), \(G\) is the number of striped socks (the thing we’re counting - so 4), and \(n\) is the number of socks you pull out (3).

\[P(X = k) = \dfrac{\binom{G}{k}\binom{N-G}{n-k}}{\binom{N}{n}}\]

Question 11

A roulette wheel has 38 pockets, 18 of them red. You bet on red for 20 spins. \(X\) is the number of spins you win.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Binomial. The spins are independent, each has the same chance of landing on red, and \(X\) counts the wins over a fixed number of spins.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim Bin(20, 18/38)\), so \(n = 20\) and \(p = 18/38\).

3. Write an expression for the probability that you win exactly 10 of the 20 spins.

Show answer

\[P(X = 10) = \binom{20}{10}\left(\dfrac{18}{38}\right)^{10}\left(\dfrac{20}{38}\right)^{10}\]

Here, \(n\) is the number of spins (20) and \(p\) is the chance that each spin lands on red (18/38).

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]

Question 12

Peter Piper picked a peck of pickled peppers (what?), but apparently a portion of Peter’s peppers were painfully picante! In other words, Peter has a bucket 40 peppers, thinking they all were just pickled and mild, but 9 were actually extremely spicy (unfortunately, they look just like the mild ones). Peter tosses 6 peppers at random onto a slice of pizza and he eats it. \(X\) is the number of spicy peppers on that slice.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Hypergeometric. Peppers that go on the pizza don’t go back in the peck, so Peter is picking without replacement, and you count how many are spicy.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim HG(40, 9, 6)\), so \(N = 40\), \(G = 9\) and \(n = 6\).

3. Write an expression for the probability that Peter’s pizza has at least 2 spicy peppers.

Show answer

\[P(X \ge 2) = 1 - P(X = 0) - P(X = 1) = 1 - \dfrac{\binom{9}{0}\binom{31}{6}}{\binom{40}{6}} - \dfrac{\binom{9}{1}\binom{31}{5}}{\binom{40}{6}}\]

Here, \(N\) is the total number of peppers in the peck (40), \(G\) is the number of spicy peppers (the thing we’re counting - so 9), and \(n\) is the number of peppers Peter puts on his pizza (6).

\[P(X = k) = \dfrac{\binom{G}{k}\binom{N-G}{n-k}}{\binom{N}{n}}\]

Question 13

A Magic 8 Ball has 20 equally likely answers, and 10 of them are some kind of “yes”. You shake it once. \(X = 1\) if the answer is a yes and \(X = 0\) if not.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Bernoulli. There’s a single shake with two outcomes, and \(X\) is 1 for a success (a yes) and 0 otherwise.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim Bernoulli(0.5)\), so \(p = 10/20 = 0.5\).

Question 14

A bag holds 30 jelly beans, 6 of them licorice. You pour out 5 to eat. \(X\) is the number of licorice jelly beans you pour out.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Hypergeometric. Jelly beans you pour out don’t go back in the bag, so you’re drawing without replacement, and you count the licorice ones.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim HG(30, 6, 5)\), so \(N = 30\), \(G = 6\) and \(n = 5\).

3. Write an expression for the probability that you don’t pour out any licorice jelly beans.

Show answer

\[P(X = 0) = \dfrac{\binom{6}{0}\binom{24}{5}}{\binom{30}{5}}\]

Here, \(N\) is the total number of jelly beans (30), \(G\) is the number of licorice beans (the thing we’re counting - so 6), and \(n\) is the number of beans you pour out (5).

\[P(X = k) = \dfrac{\binom{G}{k}\binom{N-G}{n-k}}{\binom{N}{n}}\]

Question 15

A box holds 1 gold ticket and 4 silver tickets. You draw a ticket and put it back, 6 times in all. \(X\) is the number of times you draw the gold ticket.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Binomial. Because the ticket goes back each time, the draws are independent and each has the same chance of gold, and \(X\) counts the golds over a fixed number of draws.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim Bin(6, 0.2)\), so \(n = 6\) and \(p = 1/5 = 0.2\).

3. Write an expression for the probability that you draw the gold ticket at most once.

Show answer

\[P(X \le 1) = P(X = 0) + P(X = 1) = (0.8)^6 + \binom{6}{1}(0.2)(0.8)^5\]

Here, \(n\) is the number of draws (6) and \(p\) is the chance that each draw is the gold ticket (1/5 = 0.2).

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]

Question 16

A random number app picks a whole number from 1 to 500, with every number equally likely. \(X\) is the number it picks.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Discrete uniform. Every number is equally likely, and the possible values are 1, 2, 3, and so on.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim\) Discrete Uniform\((500)\), so \(n = 500\).

Question 17

A tiny clown car pulls into the circus ring carrying 20 clowns, 8 of whom have squeaky red noses. The clowns climb out one at a time, in random order. \(X\) is the number of squeaky-nosed clowns among the first 5 to climb out.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Hypergeometric. A clown who has climbed out can’t climb out again, so the first 5 clowns are drawn without replacement, and you count the squeaky-nosed ones.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim HG(20, 8, 5)\), so \(N = 20\), \(G = 8\) and \(n = 5\).

3. Write an expression for the probability that all of the first 5 clowns out have squeaky noses.

Show answer

\[P(X = 5) = \dfrac{\binom{8}{5}\binom{12}{0}}{\binom{20}{5}}\]

Here, \(N\) is the total number of clowns (20), \(G\) is the number of squeaky-nosed clowns (the thing we’re counting - so 8), and \(n\) is the number of clowns that climb out first (5).

\[P(X = k) = \dfrac{\binom{G}{k}\binom{N-G}{n-k}}{\binom{N}{n}}\]

Question 18

Each scratch-off lottery ticket wins a prize with probability 0.1, independently of the others. You buy 8 tickets. \(X\) is the number of winning tickets.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Binomial. The tickets are independent, each has the same chance of winning, and \(X\) counts the winners among a fixed number of tickets.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim Bin(8, 0.1)\), so \(n = 8\) and \(p = 0.1\).

3. Write an expression for the probability that at least 2 of your tickets win.

Show answer

\[P(X \ge 2) = 1 - P(X = 0) - P(X = 1) = 1 - (0.9)^8 - \binom{8}{1}(0.1)(0.9)^7\]

Here, \(n\) is the number of tickets you buy (8) and \(p\) is the chance that each ticket wins (0.1).

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]

Question 19

Merlinda, a wizard in training, has learned a spell that turns a toad into a prince. Her teacher warns her that the spell only works 15% of the time. She tries it once on Gerald, the class toad. \(X = 1\) if Gerald becomes a prince and \(X = 0\) if he stays a toad.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Bernoulli. There’s a single try with two outcomes, and \(X\) is 1 for a success (a prince) and 0 otherwise.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim Bernoulli(0.15)\), so \(p = 0.15\).

Question 20

Researchers at the Institute for Extremely Serious Bird Science want to know whether Professor Pecks, a pigeon, can tell paintings by Monet from paintings by Picasso. In each trial, Professor Pecks sees one painting and pecks one of two buttons. He picks the right button 70% of the time, independently from trial to trial. \(X\) is the number of trials he gets right out of 25.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Binomial. The trials are independent, each has the same chance of a right answer, and \(X\) counts the right answers over a fixed number of trials.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim Bin(25, 0.7)\), so \(n = 25\) and \(p = 0.7\).

3. Write an expression for the probability that Professor Pecks gets all 25 trials right.

Show answer

\[P(X = 25) = (0.7)^{25}\]

Here, \(n\) is the number of trials (25) and \(p\) is the chance that Professor Pecks gets each trial right (0.7).

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]

Question 21

A raffle sells 100 tickets, and you hold 5 of them. Three different winning tickets are drawn. \(X\) is the number of your tickets that win.

1. Name the distribution of \(X\) (Bernoulli, discrete uniform, binomial or hypergeometric).

Show answer

Hypergeometric. The winning tickets are drawn without replacement, and you count how many of them are yours.

2. Give the value(s) of its parameter(s).

Show answer

\(X \sim HG(100, 5, 3)\), so \(N = 100\), \(G = 5\) and \(n = 3\).

3. Write an expression for the probability that exactly one of your tickets wins.

Show answer

\[P(X = 1) = \dfrac{\binom{5}{1}\binom{95}{2}}{\binom{100}{3}}\]

Here, \(N\) is the total number of tickets sold (100), \(G\) is the number of tickets you hold (the thing we’re counting - so 5), and \(n\) is the number of winning tickets drawn (3).

\[P(X = k) = \dfrac{\binom{G}{k}\binom{N-G}{n-k}}{\binom{N}{n}}\]