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Hard Probability Paradox: Expected Number of Die Rolls to Get a Six, Given All Rolls are Even

• 0:00:06 Introduction: The video is inspired by a "Wrath of Math" video about an "insane probability paradox." The core problem is: What's the expected number of die rolls to get a 6, given that all rolls are even numbers (2, 4, or 6)?

• 0:00:40 Initial Misconception: Many people incorrectly guess the expected number of rolls is 3, assuming a 1/3 probability of rolling a 6 on each throw with only even numbers allowed.

• 0:00:58 Paradox Explained: The presenter highlights a perceived paradox in calculating this conditional probability.

• 0:01:13 Initial Probablity Calculation: The probablity of rolling only even numbers with a six the last number rolled.

• 0:01:48 Sample Space Adjustment: The sample space is redefined to include only sequences of even numbers ending in a 6. Examples include: {6}, {2,6}, {4,6}, {2,2,6}, {2,4,6}, {4,2,6}, {4,4,6}, and so on.

• 0:02:56 Calculating Probabilities: Probabilities are calculated for each sequence in the new sample space. For example, P({6}) = 1/6, P({2,6} or {4, 6}) = (1/3) * (1/6).

• 0:03:37 Summing Probabilities (Geometric Series): The probabilities of all possible sequences are summed, forming a geometric series: (1/6) * [1 + (2/3) + (2/3)^2 + (2/3)^3 + ...].

• 0:04:03 Geometric Series Formula: The sum of the infinite geometric series is simplified to 1/6 * [1 / (1 - 2/3)] = 1/6 * 3 = 1/2. However, the presenter makes a mistake, it simplifies to 1/4.

• 0:04:18 Experiment: An experiment of many trials is conducted. The presenter keeps only the trials where all numbers rolled are even.

• 0:05:03 Conditional Probability: The conditional probability P(first roll is 6 | all rolls are even) is calculated. The numerator is the probability of sequences starting with 2 or 4 and ending with 6. The denominator is the probability of all rolls being even (previously calculated as 1/4).

• 0:05:46 Final Expression: The final expression for the conditional probability is derived, which can be used to calculate the expected number of rolls.

• 0:05:59 Intuition vs. Calculation: The presenter admits their initial intuition clashed with the mathematical result, highlighting the paradoxical nature of the problem.

• 0:06:13 Intuition: The thought process of the incorrect guess.

• 0:06:56 Simulation (Monte Carlo): The presenter mentions a computer simulation (likely using a Monte Carlo method) that confirms the theoretical calculation, validating the derived expression.

• 0:07:19 Experiment Details: The experiment is detailed to run 60,000 trials.

• 0:07:54 Unconditional Experiment: The simulation shows results for the unconditional experiment (rolling until a 6 appears, regardless of other numbers). The theoretical expected value is 6, and the simulation closely matches this.

• 0:08:11 Conditional Experiment Results: Out of 60,000 trials, approximately 1/4 (14,899) had all even numbers. This aligns with the theoretical probability of 1/4.

• 0:08:31 Expected Value Calculation: The expected number of rolls in the conditional case (all rolls even) is shown to be 4.5 in the theoretical model, the simulation also converges to 4.5.

• 0:09:17 Conclusion: The presenter concludes, reinforcing the counterintuitive result and the resolution of the paradox.

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