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
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
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
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
Each week you get multiple attempts to take a two-question quiz. For each attempt, two questions are pulled at random from a bank of 100 questions. For a single attempt, the two questions are distinct.
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Posted in Conditioning, Counting, Multiplication rule
Assume that each monkey has a strong preference between red, green, and blue M&M’s. Further, assume that the possible orderings of the preferences are equally distributed in the population. That is to say that each of the 6 possible orderings ( R>G>B or R>B>G or B>R>G or B>G>R or G>B>R or G>R>B) are found with equal frequency in the population. Lastly assume that when presented with two M&Ms of different colors they always eat the M&M with the color they prefer.
In an experiment, a random monkey is chosen from the population and presented with a Red and a Green M&M. In the first round, the monkey eats the one based on their personal preference between the colors. The remaining M&M is left on the table and a Blue M&M is added so that there are again two M&M’s on the table. In the second round, the monkey again chooses to eat one of the M&M’s based on their color preference.
[Mattingly 2022]
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Posted in Conditioning
Tagged JCM_math230_HW9_F22
Suppose that I have two coins in my pocket. One ordinary, fair coin and one coin which has heads on both sides. I pick a random coin out of my pocket, throw it, and it comes up heads.
[ Modified version of Meester, ex 1.7.35]
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Posted in Bayes Theorem, Coin Flips, Conditioning
Let \(A\) and \(B\) be two events with positive probability. When does \(\mathbf{P}(A|B)=\mathbf{P}(B|A)\) ?
Suppose that there are two boxes, labeled odd and even. The odd box contains three balls numbered 1,3,5 and the even box contains two balls labeled 2,4. One of the boxes is picked randomly by tossing a fair coin.
[Pitman p 37, example 5]
Posted in Basic probability, Coin Flips, Conditioning, Drawing Balls
At the London station there are three pay phones which accept 20p coins. one never works, another works, while the third works with probability 1/2. On my way to London for the day, I wish to identify the reliable phone, so that I can use it on my return. The station is empty and I have just three 20p coins. I try one phone and it doesn’t work. I try another twice in succession and it works both times. What is the probability that this second phone is the reliable one ?
[Suhov and Kelbert, p.10, problem 1.9]
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Posted in Bayes Theorem, Conditioning
Tagged JCM_math230_HW2_F22, JCM_math230_HW2_S13, JCM_math230_HW2_S15, JCM_math340_HW2_F13
In a certain population of people 5% have a disease. Bob’s road side clinic use a test for the disease which has a 97% of (correctly) returning positive if one has the disease and a 25% chance of (incorrectly) returning a positive if one doesn’t have the disease. If a random person is given the test, what is the chance that the result is positive ?
Now let \(\alpha\) be the chance the test returns a positive if one doesn’t have the disease. (Leave the chance that the test correctly returns a positive is one has the disease at 97%). For what value of \(\alpha\) is the chance the test is correct equal to 5% for a randomly chosen person ?
Posted in Conditioning
Consider the following model:
\(X_1,…,X_n \stackrel{iid}{\sim} f(x), \quad Y_i = \theta X_i + \varepsilon_i, \quad \varepsilon_i \stackrel{iid}{\sim} \mbox{N}(0,\sigma^2).\)
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Posted in Conditioning
Let \(X\) be the number of patients in a clinical trial with a successful outcome. Let \(P\) be the probability of success for an individual patient. We assume before the trial begins that \(P\) is unifom on \([0,1]\). Compute
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Posted in Bayes Theorem, Conditioning, Testing
An urn contains 1 black and 2 white balls. One ball is drawn at random and its color noted. The ball is replaced in the urn, together with an additional ball of its color. There are now four balls in the urn. Again, one ball is drawn at random from the urn, then replaced along with an additional ball of its color. The process continues in this way.
[From pitman p 408, #6]
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Posted in Conditioning, Urns
Tagged JCM_math230_HW10_S15, JCM_math230_HW12_F22, JCM_math230_HW9_S13, JCM_math340_HW5_F13
Consider the following hierarchical random variable
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Posted in Conditional Expectation, Conditioning, Expectations, Prior/Posterior Distribution
Tagged JCM_math230_HW10_S15, JCM_math230_HW11_F22, JCM_math230_HW9_S13, JCM_math340_HW5_F13
Consider the following mixture distribution.
What is \(\mathbf{E}(Y)\) ?. (*) What is \(\mathbf{E}(Y | X )\) ?.
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Posted in Conditioning, Expectations
Tagged JCM_math230_HW11_S15, JCM_math230_HW12_F22, JCM_math230_HW9_S13
Stark’s Pond contains 10 trout and 5 bluegill fish. Kyle catches a random number of fish (call the number \(X\)), where \(X \sim \text{Unif}(\{1,\ldots,4\})\). Once caught, that fish is removed from the pond and cannot be caught again. Each new fish comes uniformly from the remaining fish.
(a) What is the chance that Kyle catches all trout?
(b) Suppose all the fish that Kyle caught were trout. Given this information, what is the probability that he caught exactly 5 fish?
[Author Mark Huber. Licensed under Creative Commons.]
Posted in Conditioning
Consider two draws from a box with replacement contain 1 red ball and 3 blue balls. Let \(X\) be number of red balls. Let \(Y\) be 1 if the two balls are the same color and 0 otherwise. Let \(Z_i\) be the random variable which returns 1 if the \(i\)-th ball is red.
Posted in Algebra of events, Conditioning, Independence
The following is a hierarchical model.
What is \(\mathbf{E}(Y)\) ?
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Posted in Conditioning, Poisson
A digital communications system consists of a transmitter and a receiver. During each short transmission interval the transmitter sends a signal which is interpreted as a zero, or it sends a different signal which is to be interpreted as a one. At the end of each interval, the receiver makes its best guess at what is transmitted. Consider the events:
\(T_0 = \{\mbox{Transmitter sends } 0\}, \quad T_1 = \{\mbox{Transmitter sends } 1\} \)
\(R_0 = \{\mbox{Receiver perceives } 0\}, \quad R_1 = \{\mbox{Reviver perceives } 1\} \)
Assume that \(\mathbf{P}(R_0 \mid T_0)=.99\), \(\mathbf{P}(R_1 \mid T_1)=.98\) and \(\mathbf{P}(T_1)=.5\).
[Pitman page 54, problem 4]
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Posted in Bayes Theorem, Conditioning
A club contains 100 members; 51 are Democrats (or caucus with
Democrats) and 49 are Republicans. A committee of 10 members is
chosen at random.
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Posted in Binomial, Conditioning
Posted in Bayes Theorem, Conditioning
Tagged JCM_math230_HW2_F22, JCM_math230_HW2_S13, JCM_math230_HW2_S15
An insurance company has 50% urban and 50% rural customers. If every year each urban customer has an accident with probability \(\mu\) and each rural customer has an accident with probability \(\lambda\). Assume that the chance of an accident is independent from year to year and from customer to costumer. This is another way to say, conditioned on being and urban or rural the chance of having an accident each year is independent.
A costumer is randomly chosen. Let \(A_n\) be the chance this customer has an accident in year \(n\). Let \(U\) denote the event that this costumer is urban and \(R\) the event that the customer is rural.
Mathematicians and politicians throughout history have dueled.
Alexander Hamilton and Aaron Burr dueled.
The French mathematician Evariste Galois died in a duel.
Consider two individuals (H) and (B) for example dueling.
In each round they simultaneously shoot the other and the probability
of a fatal shot is \(0 < p < 1\).
1) What is the probability they are fatally injured in the same round ?
2) What is the probability that (B) will be fatally injured before (H) ?
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Posted in Coin Flips, Conditioning, Sequence of independent trials
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