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 \(\mathbb{E}(S)\).
Suppose that three fair 6-sided dice are rolled.
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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.
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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.)
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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.
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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.
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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
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.
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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.
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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
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.
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Posted in Bayes Theorem, Binomial, Confidence Interval
Three boxes contain yellow and green balls
One box is selected at random, and one ball is pulled out of that box.
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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.
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.
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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 red cards (numbered 1 through 20)
Four cards are dealt without replacement from this deck.
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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:
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, \(p=\mathbb{P}(Heads).\)
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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:
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.
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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.
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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.
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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