Your overall probability of winning something is 1/3. Your probability of winning the current given round improves each round (1/12, 1/11, 1/10, 1/9, respectively). Because the probability of winning the second round only applies if you didn't win the first round, and your probability of winning the third round only applies if you lost the first and second, and so on and so forth for successive rounds, to convert individual probabilities to total probability requires you to recursively multiply successive terms. You must add together each individual probability multiplied by the probability of losing the previous rounds.
Note: 12/12 is your chance of losing round 0, which doesn't exist. Therefore your probability of losing that round is 1.
Probability of winning round 1 + round 2 + round 3 + round 4 = Total probability
More likely than a Calculus teacher lacking a grasp of basic probability theory is that your friend was only looking to hear what they wanted to hear. When pressed, the teacher validated their assertion that "the chances get better with each round," and that was enough.
Note: 12/12 is your chance of losing round 0, which doesn't exist. Therefore your probability of losing that round is 1.
Probability of winning round 1 + round 2 + round 3 + round 4 = Total probability
(1/12 * 12/12) + (1/11 * 12/12 * 11/12) + (1/10 * 12/12 * 11/12 * 10/11) + (1/9 * 12/12 * 11/12 * 10/11 * 9/10) = 1/3 = 0.333333 ( ... )
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