Probability of At Least One: 3 Missiles at 70% Each = 210%?

Step by step

1. Each missile: 70% chance to hit
   Each missile hits its target seventy percent of the time.
2. Fire 3: 70% + 70% + 70% = 210%?
   Fire three at the same target. So that's two hundred ten percent, right?
3. 2022: a retired general said it on TV
   A retired Taiwanese general, Yu Beichen, said exactly this on TV in 2022.
4. Later: a Chinese high school exam question
   A few years later, a high school exam in China turned it into a test question.
5. A 70% · B 97.3% · C 100% · D 210%?
   So which would you pick? Seventy, ninety-seven point three, a hundred, or two ten?
6. A probability can't go over 100%
   First, two ten can't be right. A probability can never go over a hundred percent.
7. Way 1: each missile misses 30%
   Way one: flip the question. Each missile misses thirty percent of the time.
8. All 3 miss: 0.3 × 0.3 × 0.3
   What's the chance all three miss? Point three, times point three, times point three...
9. = 0.027 = 2.7%
   two point seven percent.
10. At least one hits: 100% − 2.7%
    So at least one hits: a hundred, minus two point seven...
11. = 97.3%
    ninety-seven point three percent.
12. Way 2: 1,000 attacks · missile 1 stops 700
    Way two: picture a thousand attacks. The first missile stops seven hundred. Three hundred get through.
13. Missile 2 stops 70% of 300 = 210
    The second missile stops seventy percent of those three hundred: two hundred ten.
14. There's the 210! 210 attacks, not 210%
    Wait. There's the two ten! But it's two hundred ten attacks, not two hundred ten percent.
15. Missile 3 stops 63 of 90 · 27 get through
    Ninety get through. The third missile stops sixty-three of them. Twenty-seven get through.
16. 973 out of 1,000 = 97.3%
    Nine hundred seventy-three out of a thousand. Ninety-seven point three percent.
17. Only if the missiles don't affect each other: independent
    One catch: this only works if the missiles don't affect each other. That's called independent.
18. On a test: 1 − P(all miss)
    On a test: each missile misses with probability point three. They're independent, so all three miss with probability point three cubed, point zero two seven. So at least one hits with probability one minus point zero two seven, which is point nine seven three.
19. On a test, it looks like this
    On a test, it can hide like this.