概率

考什么 · What it tests

需要先会 · Prerequisites

• 基本计数原理（乘法/加法） / Basic counting (multiplication/addition)
• 分数与约分 / Fractions and simplifying
• 排列与组合 / Permutations and combinations

常见套路 · Common moves

1. 先把样本空间数清：分母（所有等可能结果）数对了，概率题就成了一半——这是一切概率题的地基。

   Nail down the sample space first: get the denominator (all equally-likely outcomes) right and the problem is half solved — the foundation of every probability question.
2. 「至少」用 1 减：P(\text{至少一次}) = 1 - P(\text{一次都没有})，正面枚举麻烦时反面往往一步出。

   Handle "at least" by subtracting from 1: $P(\text{at least once}) = 1 - P(\text{never})$ — when the direct count is messy, the complement often finishes in one step.
3. 几何概型算面积比 / 长度比：结果连续（如落点、时刻）时，概率 = 有利区域 ÷ 总区域的面积或长度之比。

   For geometric probability, use area or length ratios: when outcomes are continuous (a landing spot, a moment in time), the probability is the favorable region over the whole region.

易错点 · Common pitfalls

• 结果不等可能却当等可能：两个骰子和为 7 的方式比和为 2 多，直接按「和有 11 种」算分母就错了。

  Treating unequal outcomes as equal: two dice make a sum of 7 in more ways than a sum of 2, so using "11 possible sums" as the denominator is wrong.
• 有放回和无放回混：抽完放回去分母不变，不放回时第二次的分母要减 1。

  Mixing with- and without-replacement: putting the item back keeps the denominator fixed, but without replacement the second draw's denominator drops by 1.