Suppose that three fair 6-sided dice are rolled.

- Let \(M\) be the minimum of three numbers rolled. Find \(\mathbb{E}(M)\).
- Let \(S\) be the sum of the largest two rolls. Find \(\mathbb{E}(S)\).

Learning probability by doing !

Suppose that three fair 6-sided dice are rolled.

- Let \(M\) be the minimum of three numbers rolled. Find \(\mathbb{E}(M)\).
- Let \(S\) be the sum of the largest two rolls. Find \(\mathbb{E}(S)\).

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Posted in Dice Rolls, Expectations, Max and Mins, Tail Sum Fromula

A host invites \(n\) guests to a party (guest #1, guest #2, … , guest #n). Each guest brings with them their best friend. At the party there is a large circular table with \2n\) seats. All of the \(n\) invited guests and their best friends sit in a random seat.

- What is the probability that guest #1 is seated next to their best friend?
- What is the expected number of the \(n\) invited guests who are seated next to their best friend?

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Posted in Expectations, Indicator functions

Telephone calls come in to a customer service hotline. The number of calls that arrive within a certain time frame follows a Poisson distribution. The average number of calls per hour depends on the day of the week. During the week (Monday through Friday) the hotline receives an average of 10 calls per hour. Over the weekend (Saturday and Sunday) the hotline receives and average of 5 calls per hour. The hotline operates for 8 hours each day of the week. (The number of calls on one day is independent of the numbers of calls on other days.)

- What is the probability that the center receives more than 500 calls in 1 week?
- Each person who calls the center has a 20% chance of getting a refund (independent of other callers). Find the probability that 10 or fewer people get a refund on Tuesday.
- One day of the week is chosen uniformly at random. On this day, a representative at the call center reports that 60 people called in. Based on that information, what is the probability that the day was a weekend day (either Saturday or Sunday)?

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Posted in Bayes Theorem, Poisson

15 players each roll a fair 6-sided die once. If two or more players roll the same number, those players are eliminated. What is the expected number of players who get eliminated?

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Posted in Expectations, Indicator functions

Calls arrive at a call center according to a Poisson arrival process with an average rate of 2 calls/minute. Each caller has a 1/12 chance of having a January birthday, independent of other callers. What is the expected wait time until the call center receives 3 calls from callers with January birthdays?

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Posted in Expectations, Poisson arrivial process

A game of rock paper scissors consists of several rounds (players continue to play rounds until one player wins). In one round of rock paper scissors, two players each choose one of three options (rock, paper, or scissors). If they choose two different options, the game ends (rock beats scissors, scissors beat paper, and paper beats rock). If both players choose the same option, the game continues for another round. Assume each player chooses rock, paper or scissors uniformly at random and independently.

- What is the distribution of number of rounds played in a single game?
- What is the expected number of rounds in a game? What is the standard deviation of number of rounds in a game?
- Let \(n\) be the number of games played. How big must \(n\) be to ensure at least 100 rounds are played with 90% probability? Use an appropriate approximation to estimate.

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Posted in Geometric Distribution, Mean and Variance, Normal/CLT approximation

Birds arrive at a bird feeder according to a Poisson arrival process with a rate of 6 birds per hour. A person starts watching the feeder at time 0.

- What is the probability that the first three birds arrive at the feeder within 30 minutes?
- What is the expected time it takes for the 10th bird to arrive?
- 10% of the birds who visit the feeder are cardinals. What is the probability that the 3rd cardinal to arrive at the feeder is the 10th bird to arrive at the feeder?
- 10% of the birds who visit the feeder are cardinals. The person watching the feeder decides to continue watching the feeder until they see a cardinal. What is the probability that the person waits more than 5 hours?

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Posted in Negative Binomial, Poisson arrivial process

An ant is crawling on a number line. The ant starts out at position \(0\). Every second the ant either

- Moves to the right 1 unit, with probability 1/2,
- Moves to the left 1 unit, with probability 1/4, or
- Stays at its current location, with probability 1/4

The ant’s movement during a particular second is independent of the ant’s previous movements. Let \(X_{160}\) be the ant’s location after 160 seconds.

- In 160 seconds, what is the probability that the ant moves to the right exactly 80 times, and to the left exactly 40 times?
- What is \(\mathbb{E}(X_{160})\)?
- What is \(Var(X_{160})\)?
- Let \(\mu=\mathbb{E}(X)\). Estimate \(\mathbb{P}(|X-\mu|\geq 15)\).

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Posted in Mean and Variance, Multinomial, Normal/CLT approximation

A host invites guests to a party. How many guests should be invited in order for the expected number of guests who share a birthday with at least one other guest to be at least 4?

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Posted in Expectations, Indicator functions

An experimenter rolls a fair 6-sided die until they’ve seen both a 1 and a 2 (not necessarily consecutively). What is the experimenter’s expected number of rolls?

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Posted in Expectations, Geometric Distribution

Let \(X\) be a random variable with probability mass function

\(p(n) = \frac{1}{c^n}\quad \text{for } n=2,3,4,\cdots\)

and \(p(x)=0\) otherwise.

- Find \(c.\)
- Compute the probability that \(X\) is even.

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Posted in probability mass function, Series

An ant crawls along a coordinate grid. The ant starts at \((0,0)\). At each step, the ant either moves up one unit (with probability 1/2) or to the right 1 unit (with probability 1/2).

After 5 steps the ant has

- a 5/32 chance of being at the coordinate \((4,1)\),
- a 10/32 chance of being at the coordinate \((3,2)\), and
- a 17/32 chance of being at one of the coordinates \((2,3), (1,4), (5,0), (0,5)\)

What is the probability that the ant is at position \((4,2)\) after 6 steps?

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Posted in Conditioning

A magician claims to have a magic die. If the die is rolled and lands on an even number, then the next time the die is rolled it will land on an odd number (and vice versa). So, as the die is rolled it will alternate perfectly between even and odd numbers (or so the magician claims).

You, being skeptical, figure there’s a 1 percent chance that the die is magical and a 99 percent chance that it’s just an ordinary fair die. You then ask the magician to “prove” the die is magical by rolling it some number of times.

How many successfully alternating rolls will it take for you to think there’s a 99 percent chance the die is magical (or, more likely, that it’s rigged in some way so it always alternates)?

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Posted in Bayes Theorem, Sequence of independent trials, Series

Tagged Modified from fivethirtyeight: https://fivethirtyeight.com/features/can-you-flip-the-magic-coin/

You have a fair coin and a biased coin, but you can’t tell which is which. The biased coin lands on heads 75% of the time. You decide to try to determine which coin is the biased coin by selecting one of the coins at random and flipping tn 100 times. Let \(\hat{p}\) be your observed fraction of heads. Based on \(\hat{p}\), you decide which coin is the biased one.

- For which values of \(\hat{p}\) will you assume the coin you flipped is the biased coin?
- What is the probability that you correctly determine which coin is the biased coin?

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Posted in Bayes Theorem, Binomial, Confidence Interval

Three boxes contain yellow and green balls

- Box 1 contains 2 yellow balls.
- Box 2 contains 2 green balls.
- Box 3 contains 1 yellow ball and 1 green ball.

One box is selected at random, and one ball is pulled out of that box.

- The ball that is pulled out of the chosen box is yellow. What is the probability that the other ball in that same box is also yellow?
- Let \(A\) be the event that Box 3 is chosen. Let \(B\) be the event that a yellow ball is pulled out of the chosen box. Are \(A\) and \(B\) independent?
- The ball that is pulled out of the chosen box is yellow. Without replacement, a second ball is chosen at random from one of the three boxes. (Each box has a 1/3 chance of being selected.) What is the probability that the second ball chosen is also yellow?

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Posted in Bayes Theorem, Drawing Balls, Drawing without replacement, Independence

Let \(A_n\) be the event that in \(n\) flips of a fair coin, there are never 2 consecutive tails. Suppose we know the following probabilities.

- \(\mathbf{P}(A_{19})\approx 0.021\)
- \(\mathbf{P}(A_{20})\approx 0.017\)

Evaluate \(\mathbf{P}(A_{21})\)

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Posted in Coin Flips, Conditioning

Consider the following joint distribution.

If the experiment is flipping a fair coin three times, which of the following could be the random variables \(X\) and \(Y\). Select all that apply.

- \(X=\) the number of heads, \(Y=\) the number of tails.
- \(X=\) the number of tails, \(Y=\) the number of tails (i.e., \(Y=X\)).
- \(X=\) the number of heads. \(Y=3-X.\)
- \(X=\) the number of tails on the first two flips. \(Y=\) the number of tails on the last two flips.

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Posted in Joint Distributions

About 60% of the world’s population has brown eyes. About 20% of the world’s population has brown hair. Given that a person has brown eyes, they have a 10% chance of also having brown hair.

Given that a randomly selected person does not have brown eyes what is the probability that they also do not have brown hair?

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Posted in Conditioning

In a non-standard deck of cards there are

- 20 blue cards (numbered 1 through 20),
- 20 green cards (numbered 1 through 20), and

20 red cards (numbered 1 through 20)

Four cards are dealt without replacement from this deck.

- What is the probability that exactly two of the four cards dealt are blue?
- Given that at least one of the first two cards dealt is blue, what is the probability that exactly three of the four cards dealt are blue?
- What is the probability that at least two of the four cards dealt have the same numeric value (1 through 20)?

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Posted in Cards, Drawing without replacement, Multiplication rule

An experimenter has two fair coins and one biased coin. The biased coin lands on heads with probability 3/4.

The experimenter randomly selects one of the three coins and flips it until they get heads.

Let \(A\) be the event that the experimenter flipped the biased coin.

Let \(B\) be the event that it took the experimenter an even number of flips to get heads.

Are events \(A\) and \(B\) independent?

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Posted in Coin Flips, Geometric Distribution, Independence, Series

A warehouse stores batteries. Most of the batteries work properly, but about 0.1%$ are faulty.

If a company orders 500 batteries, what is the probability that less than 3 will be faulty? Do this problem three ways:

- Find the probability exactly.
- Use a Poisson approximation to estimate.
- Use a normal approximation to estimate.

A company needs 10,000 working batteries. How many batteries should the company order from the warehouse in order to be 99.7% certain that they will receive at least 10,000 working batteries?

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Posted in Binomial, Normal/CLT approximation, Possion approximation

You have a biased coin, but you don’t know what the bias is. Let \(p\) be the actual probability of getting heads on a single coin flip, \(p=\mathbb{P}(Heads).\)

- Suppose \(p=0.8\). What is the probability of observing between 76 and 84 heads out of 100 flips of the coin.
- Suppose you flip the coin 100 times and observe 80 heads. What is the 95% confidence interval for \(p\)?

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Posted in Binomial, Coin Flips, Confidence Interval, Normal/CLT approximation

About 9% of birthdays (in the US) are in August. A researcher samples 10,000 people from the US and asks for their birthdays. Estimate the probability that between 850 and 950 of those people were born in August.

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Posted in Binomial, Normal/CLT approximation

Suppose you have three boxes, \(Box_1,Box_2,Box_3\), such that \(Box_i\) contains \(i\) white balls and one black ball.

You will to select one ball from the boxes. Here are two schemes you could use for selection:

- Select one box uniformly at random. Pull one ball from that box. Or,
- Dump all the balls into one box. Mix them up. Pull out one ball.

Are these two schemes probabilistically equivalent?

Suppose instead of selecting a box uniformly at random, you select \(Box_i\) with probability \(p_i\). Find a list of values for \(p_1, p_2,\) and \(p_3\) that would make this new scheme probabilistically equivalent to scheme 2?

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Posted in Bayes Theorem, Drawing Balls

Consider a binomial\((10,p)\) distribution. If \(p\) is chosen uniformly at random from the interval \((0,1)\), what is the likelihood that the most likely number of the binomial distribution will be less than the mean of the binomial distribution?

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Posted in Binomial, Mean and Variance, Uniform

Over his career, Shaquille O’Neal made about 53% of his free throws. Assume his probability of making a single free throw is 53%. Suppose Shaq shot a round of 20 free throws and you’re told he made 15 of them.

- What is the likelihood he made the first free throw, given that he made 15?
- What is the likelihood he made at least 1 out of his first 5 free throws, given that he made 15?

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Posted in Binomial, Conditioning

You have a pair of fair dice and a pair of loaded dice. But you forgot which pair is which. You do remember that when you bought the loaded dice, the company that makes them claimed the dice would land on a sum of 7 approximately 1/3 of the time.

- You choose one of the pairs at random and roll it once. You get a sum of 7. What is the likelihood that you picked the loaded dice?
- You choose one of the pairs at random and roll the pair three times. You get exactly one sum of 7. What is the likelihood that you picked the loaded dice?

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Posted in Bayes Theorem, Binomial, Conditioning, Dice Rolls

Let \(\Omega\) be an outcome space with 16 outcomes. \(A\) and \(B\) are events inside of \(\Omega\). Event \(A\) has 10 outcomes and event \(B\) has 10 outcomes.

- Determine all the possible values of \(\# (A\cap B).\)
- Determine all the possible values of \(\# (A\cup B).\)
- Determine all the possible values of \(\#(A^c\cup B^c).\)
- Determine all the possible values of \(\# (A^c\cap B^c).\)

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Posted in Algebra of events, Counting

You roll a fair 6-sided die 3 times. What is the likelihood of getting exactly one 4, exactly one 5, or exactly one 6?

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Posted in Addition rule, Counting, Dice Rolls

A researcher is collecting data from 10 high school classrooms. Each classroom contains 30 people. The researcher asks each student to fill out a survey. Suppose each student has about a 40% chance of completing the survey (independent of other students). What is the probability that at least 4 classrooms have at least 15 students who complete the survey?

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Posted in Binomial