钥匙在左口袋的概率

带钥匙的概率为 $0.8$，且若带了钥匙，在右口袋概率 $0.5$。

“右口袋没有”的概率由两种情况组成：

• 根本没带钥匙：$0.2$；
• 带了钥匙但在左口袋：$0.8\times 0.5=0.4$。

因此

$$P(\text{左}\mid\text{右无})=\frac{0.4}{0.2+0.4}=\frac{2}{3}.$$

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Original Explanation

• Probability keys are indeed on you is $0.8$. Then if on you, chance they are in right pocket is $0.5$.
• You check right, not found => either you didn’t bring them at all (20%) or they are in left pocket (40%). So total “not in right” is $0.2 +0.4=0.6.
  Thus $$P(\text{left}|\text{not right}) = \frac{0.4}{0.6} = \frac{2}{3}.$$