← Five Rings Interview Insights
This one tripped me up more than it should have.
Model the problem as a Markov chain with states representing the longest suffix of the current sequence that matches a prefix of either pattern. Set up equations for the probability of HHH winning from each state and solve them, or use a symmetry argument to simplify. Clearly explain the state definitions and the transitions.
Pro tip: Mention that this is a classic example of Penney's game and that the second pattern (THH) has a higher probability of appearing first (3/4 vs 1/4) due to overlapping self-transitions in HHH. This shows awareness of pattern matching and non-intuitive results.
Identify the two patterns: HHH and THH. Define states based on the longest suffix of the current sequence that is a prefix of either pattern. For example, states can be: start (no progress), H, HH, HHH (absorbing), T, TH, THH (absorbing).
Let p_i be the probability that HHH appears before THH starting from state i. Write equations for each non-absorbing state based on the next coin flip. For absorbing states, p_HHH = 1 and p_THH = 0.
Solve the linear system for p_start. Alternatively, use a symmetry argument: note that THH can only occur after a T, and once a T occurs, HHH cannot occur until the sequence resets. This leads to a simpler calculation.
Check the result using the Conway leading number algorithm or by considering the expected waiting times. This ensures the answer is correct and demonstrates depth.
State the final probability (3/4 for THH, 1/4 for HHH) and explain why it makes intuitive sense: HHH requires three consecutive heads, and any tail resets progress, while THH can be achieved after a tail followed by two heads.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by clarifying that the question asks for the expected value of the maximum run length of heads in 15 fair coin tosses. Then, outline a method to compute it, either by deriving the distribution of the longest run using dynamic programming or by using known approximations and verifying with simulation. Finally, present the numerical answer with appropriate reasoning.
Pro tip: Show that you can connect theory to practice: mention that in real-world data science, you'd often simulate such problems to validate analytical results, and discuss how this relates to streak detection in user behavior or A/B testing.
Restate the question to ensure understanding: we need the expected value of the longest consecutive sequence of heads in 15 independent fair coin tosses.
Decide between an analytical approach (e.g., dynamic programming to compute the exact distribution) or a simulation-based estimate. For an interview, an analytical approach is preferred, but simulation can be mentioned as a sanity check.
If using dynamic programming, define states for the current run length and the maximum run so far, and compute probabilities recursively. Alternatively, use known formulas or approximations for the expected longest run.
State the expected value, e.g., approximately 3.3, and explain how you arrived at it, including any assumptions or simplifications.
Mention how this concept applies to real-world scenarios like streak analysis in user engagement or quality control, and note that simulation can be used for larger n.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Bayesian updating, which I actually felt okay about.
Recognize that the prior is a Beta(4,1) distribution and the likelihood is Binomial, so the posterior is conjugate and also Beta. Compute the posterior parameters by adding the number of successes and failures to the prior parameters, yielding Beta(8,1).
Pro tip: Emphasize that the prior is equivalent to having seen 3 heads and 0 tails, which shows you understand how to interpret prior parameters in a Bayesian context.
The prior density is proportional to θ^3 on [0,1], which is a Beta(4,1) distribution. Recognize that this is a conjugate prior for the binomial likelihood.
The data (4 heads in 4 tosses) follows a Binomial(4, θ) likelihood, which is proportional to θ^4 (1-θ)^0.
The posterior is proportional to the product of the prior and likelihood: θ^3 * θ^4 = θ^7. Thus, the posterior density is proportional to θ^7 on [0,1].
The kernel θ^7 corresponds to a Beta(8,1) distribution. So the posterior is Beta(8,1).
The posterior mean is 8/9, and the distribution is heavily concentrated near 1, reflecting strong evidence that θ is close to 1.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.