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Recognize this as a Bayesian inference problem where you need to update the prior probability of picking the biased coin (0.5) based on the observed data (two Heads in the first two flips). Compute the likelihood of observing two Heads under each coin (fair: 0.25, biased: 0.49) and apply Bayes' theorem to find the posterior probability.
Pro tip: State your assumptions clearly (e.g., independence of flips, uniform prior) and consider mentioning that the third flip is irrelevant to the conditional probability given the first two flips, showing you understand conditional independence.
Let B be the event of picking the biased coin, F the fair coin. Prior: P(B) = P(F) = 0.5. Let E be the event that the first two flips are Heads.
Under fair coin: P(E|F) = (0.5)^2 = 0.25. Under biased coin: P(E|B) = (0.7)^2 = 0.49.
Use P(B|E) = P(E|B)P(B) / [P(E|B)P(B) + P(E|F)P(F)] = (0.49*0.5) / (0.49*0.5 + 0.25*0.5) = 0.49 / (0.49+0.25) = 0.49/0.74 ≈ 0.662.
The probability is approximately 66.2%. Explain that observing two Heads increases the likelihood of the biased coin from 50% to about 66%.
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You have to use the posterior from part one here, not the original priors.
First, clarify the assumptions about the coin and the flipping process, especially whether the flips are independent and the coin is fair. Then, if independence holds, explain that the probability of the third flip being Heads is simply 1/2, regardless of previous outcomes. If there is any doubt, discuss how you would test for independence or update beliefs using Bayes' theorem.
Pro tip: Always state your assumptions explicitly—interviewers at BCG value clarity and rigor. Mention that in real-world data, you would check for independence and fairness before applying the simple probability.
Ask or state whether the coin is fair and whether flips are independent. This sets the foundation for the probability calculation.
If flips are independent, the outcome of the first two flips does not affect the third. Thus, the probability remains 1/2 for a fair coin.
If independence is not assumed, discuss how you would model dependence (e.g., a biased coin with unknown bias) and use Bayesian updating to compute the probability.
Explain how you would test for independence and fairness using data, such as chi-square tests or Bayesian methods, to validate assumptions.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Recognize that after observing two Heads, the process effectively restarts because the coin is memoryless. Let X be the additional flips needed to get a Tail, so T = 2 + X. Since each flip is independent with P(Tail)=p, X follows a geometric distribution with mean 1/p, hence E[T] = 2 + 1/p.
Pro tip: Explicitly state the memorylessness property and define p as the probability of Tails; this shows you understand the underlying assumption and avoids confusion if p is not 0.5.
Let T be the total number of flips until the first Tail, including the Tail flip. Given that the first two flips are Heads, we can write T = 2 + X, where X is the number of additional flips needed to get a Tail.
Because coin flips are independent, the fact that the first two flips were Heads does not affect future flips. Thus, X is the waiting time for the first Tail starting from scratch, which follows a geometric distribution.
X ~ Geometric(p), where p is the probability of getting a Tail on a single flip. The expected value of a geometric random variable is E[X] = 1/p.
Using linearity of expectation, E[T] = E[2 + X] = 2 + E[X] = 2 + 1/p. If the coin is fair (p=0.5), then E[T] = 2 + 2 = 4.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.