This is the one I second-guessed myself on.
Use linearity of expectation by decomposing the process into two stages: waiting for the first head and then waiting for the second head. Recognize that each stage follows a geometric distribution with success probability 1/2, so the expected number of flips per stage is 2, giving a total of 4. Alternatively, model the process as a Markov chain or use negative binomial distribution to confirm the result.
Pro tip: Explicitly state that the expectation is 4 and then briefly justify why the memoryless property of the geometric distribution makes the second stage identical to the first. This shows you understand the underlying probabilistic structure and can communicate it clearly.
Let X be the total number of flips needed to get two heads. Break X into X1 (flips to get first head) and X2 (additional flips to get second head after the first).
Recognize that X1 and X2 are independent and each follows a geometric distribution with success probability p = 1/2, so E[X1] = E[X2] = 1/p = 2.
Since X = X1 + X2, the expected total flips is E[X] = E[X1] + E[X2] = 2 + 2 = 4.
If time permits, mention that the negative binomial distribution gives the same result: expected flips for r=2 successes is r/p = 2/(1/2) = 4.
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Start by clearly defining the sample space for three fair coin flips, then count the number of outcomes with exactly two heads. Use the binomial probability formula to compute the probability, and explain each step to show your reasoning.
Pro tip: After giving the answer, briefly mention how this simple calculation generalizes to the binomial distribution, which is fundamental in many data science applications such as A/B testing and classification. This shows you can connect basic probability to practical data science work.
State that each flip has two equally likely outcomes (H or T), and for three flips there are 2^3 = 8 possible sequences. List them if helpful.
Count the sequences with exactly two heads: HHT, HTH, THH. There are 3 such outcomes.
Divide the number of favorable outcomes by the total number of outcomes: 3/8 = 0.375.
Use the binomial probability formula: P(X=2) = C(3,2) * (0.5)^2 * (0.5)^1 = 3 * 0.25 * 0.5 = 0.375, confirming the result.
Explain that the probability is 37.5%, and note that this is a specific case of the binomial distribution with n=3, p=0.5.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
State the definition of conditional probability: P(A|B) = P(A and B) / P(B). Plug in the given values and compute the result. Then briefly interpret the result in the context of the problem.
Pro tip: After computing, mention that P(A|B) = 0.3 equals P(A), indicating independence. This shows deeper understanding and connects to real-world data science scenarios.
Write down the definition of conditional probability: P(A|B) = P(A ∩ B) / P(B), assuming P(B) > 0.
Plug in P(A ∩ B) = 0.15 and P(B) = 0.5 into the formula.
Perform the division: 0.15 / 0.5 = 0.3.
Note that P(A|B) = 0.3, which equals P(A). This suggests that A and B are independent events.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Recognize that this is a straightforward application of the definition of conditional probability: P(A|B) = P(A and B) / P(B). Plug in the given values and compute the result, then briefly interpret what it means in context.
Pro tip: After computing the numerical answer, mention that since P(A|B) = P(A) = 0.3, events A and B are independent. This demonstrates deeper understanding and connects to real-world data science scenarios like feature independence in Naive Bayes.
Recall the definition of conditional probability: P(A|B) = P(A and B) / P(B), provided P(B) > 0.
Substitute P(A and B) = 0.12 and P(B) = 0.4 into the formula: P(A|B) = 0.12 / 0.4.
Perform the division: 0.12 ÷ 0.4 = 0.3. So P(A|B) = 0.3.
Note that P(A|B) equals P(A) = 0.3, which indicates that A and B are independent events. This means knowing B occurred does not change the probability of A.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.