The single-atom survival probability is just (1/2)^10, which is clean enough.
Break the problem into four parts: compute the survival probability per atom using the half-life, identify the distribution of the number of survivors as Binomial(n=100, p), then compute the expected value and the probability of at least one survivor using the complement rule. Explain each step clearly and connect the concepts.
Pro tip: Emphasize that the atoms are independent and identical, which justifies the binomial model; also note that the probability at least one survives is not simply 100 times the individual survival probability due to dependence in the complement.
Use the half-life formula: after each day, the probability an atom survives is 1/2. After 10 days, the survival probability is (1/2)^10 = 1/1024 ≈ 0.0009765625.
Since there are 100 independent atoms, each with the same survival probability p, the number of surviving atoms follows a Binomial distribution with parameters n=100 and p=(1/2)^10.
For a binomial distribution, the expected value is n*p = 100 * (1/1024) = 100/1024 ≈ 0.09765625.
Use the complement rule: P(at least one) = 1 - P(none survive) = 1 - (1 - p)^100 = 1 - (1 - 1/1024)^100. Approximate or compute exactly.
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Start by defining the events and given probabilities clearly, then apply Bayes' theorem step by step. Derive the posterior probability formula and explain each component, emphasizing the role of the base rate and test accuracy. Finally, discuss practical implications such as the impact of low prevalence on posterior probability.
Pro tip: Always sanity-check the result with extreme values (e.g., prevalence near 0 or 1) to ensure the formula behaves as expected, and mention that this is the foundation of Bayesian inference used in many data science applications.
Let A be the event that the condition is present, and B be the event of a positive test result. Define P(A) as the prior, P(B|A) as the sensitivity, and P(B|¬A) as the false positive rate.
Write Bayes' theorem: P(A|B) = P(B|A) * P(A) / P(B). Explain that P(B) is the total probability of a positive test.
Calculate P(B) = P(B|A)P(A) + P(B|¬A)P(¬A), where P(¬A) = 1 - P(A).
Plug P(B) into Bayes' theorem to get the posterior: P(A|B) = P(B|A)P(A) / [P(B|A)P(A) + P(B|¬A)(1-P(A))].
Explain what the posterior represents and discuss how it changes with different priors or test accuracies, highlighting the base rate fallacy.
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The first one is easy, slope is Cov(y,x)/Var(x) = 1.
First, clarify that the question asks for the population slopes from simple linear regressions, not sample estimates. Then, derive the slope for regressing y on x as Cov(x,y)/Var(x) and for regressing x on y as Cov(x,y)/Var(y), using the independence of x and e to compute the moments. Finally, plug in the values to get 1 and 1/2, and explain why they differ.
Pro tip: Emphasize that regression slopes are not symmetric and depend on which variable is treated as the response; this is a common pitfall in causal inference and predictive modeling.
State that you are finding the population slope coefficients for the simple linear regressions y on x and x on y, assuming the standard normal distributions for x and e.
For a simple linear regression of Y on X, the population slope is β = Cov(X,Y)/Var(X). Similarly, for X on Y, the slope is Cov(X,Y)/Var(Y).
Given y = x + e, with x and e independent standard normals, compute Cov(x,y) = Var(x) = 1, Var(y) = Var(x) + Var(e) = 2.
Plug in: slope of y on x = 1/1 = 1; slope of x on y = 1/2 = 0.5.
Explain that the slopes differ because regression is asymmetric; the slope depends on which variable is the response. Mention implications for modeling and causality.
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Switching wins with probability 2/3, staying wins with 1/3.
Start by clearly stating that you should switch, with a 2/3 winning probability, versus 1/3 if you stay. Then explain the reasoning using conditional probability or a simulation-based argument, emphasizing that the host's action provides information. Finally, connect it to how you would validate such probabilistic reasoning in a data science context, e.g., via Monte Carlo simulation.
Pro tip: Mention that the host's behavior is crucial: he always opens a losing door and always offers the switch. If the host's strategy were different, the probabilities could change. This shows you understand the assumptions behind the problem.
Begin by directly answering that you should switch, and give the probabilities: 2/3 for switching, 1/3 for staying.
Describe the intuitive explanation: initially, your chosen door has a 1/3 chance, so the other two doors collectively have 2/3. When the host reveals a losing door, that 2/3 probability collapses onto the remaining unopened door.
Use Bayes' theorem or a probability tree to show that P(win|switch) = 2/3 and P(win|stay) = 1/3, given the host always opens a losing door and offers the switch.
Clarify that the probabilities are not 50-50 because the host's action is not random; he knows where the prize is and deliberately avoids it.
Mention that this problem illustrates the importance of understanding data-generating processes and that you can verify the result via Monte Carlo simulation, a common technique in data science.
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Recognize this as the coupon collector's problem and derive the expected value by summing the expected waiting times for each new face. Use linearity of expectation, where the probability of seeing a new face when k faces remain is k/n, so the expected rolls for that phase is n/k. Sum over k=1 to n to get n * H_n.
Pro tip: Mention that this is a classic problem and that the harmonic number approximation gives n ln n + γn + 1/2, which is useful for large n. Also, note that the variance is also known and can be derived similarly, showing depth.
Recognize that this is the coupon collector's problem: collecting all n distinct faces of a fair die.
Break the process into phases: phase k starts when you have n-k distinct faces and ends when you see a new face, leaving n-k-1 faces unseen.
In phase k, the probability of seeing a new face is k/n, so the expected number of rolls is n/k.
By linearity of expectation, total expected rolls is sum_{k=1}^n n/k = n * H_n, where H_n is the n-th harmonic number.
For large n, H_n ≈ ln n + γ + 1/(2n), so expected rolls ≈ n ln n + γn + 1/2. Mention this to show practical insight.
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A probability statement treats parameters as fixed and asks how likely the data is.
Start by defining the likelihood function as a function of the parameters given fixed observed data, emphasizing that it is not a probability distribution over parameters. Then contrast it with a probability statement, which treats parameters as fixed and data as random, and highlight the inversion of roles. Use a simple example like coin flips to illustrate the difference concretely.
Pro tip: Mention that the likelihood is not a probability density over parameters, so integrating it over parameters does not yield 1—this subtle point often trips up candidates and shows deep understanding. Also, connect it to why we maximize likelihood (MLE) rather than compute probabilities of parameters.
State that the likelihood function L(θ | x) represents the plausibility of parameter values θ given observed data x, and is proportional to the probability of the data as a function of θ.
Explain that a probability statement P(x | θ) treats θ as fixed and x as random, describing the distribution of data under known parameters.
Highlight that in likelihood, data is fixed and parameters vary, while in probability, parameters are fixed and data varies. Emphasize that likelihood is not a probability distribution over θ.
Use a simple example, such as flipping a coin with unknown bias θ, to show how the likelihood of θ given observed heads/tails differs from the probability of observing heads given θ.
Explain how likelihood is used in parametric modeling for estimation (e.g., MLE) and inference, and note that it is not normalized like a probability density over parameters.
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