Inclusion–Exclusion

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| Questions: 10 | Updated: Oct 13, 2025
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1) Given P(A)=0.58, P(B)=0.46, P(A∩B)=0.31, find P(A∪B).

Explanation

P(A∪B)=0.58+0.46−0.31=0.73.

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About This Quiz
Inclusionexclusion - Quiz

Overlapping sets often confuse probability and counting problems. The inclusion–exclusion principle clears up the confusion! In this quiz, you’ll practice applying the principle to handle multiple intersections. Try this quiz to unlock one of the most powerful tools in combinatorics.

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2) |A|=240, |B|=180, |A∩B|=95, total n=500. Find P(A∪B).

Explanation

(240+180−95)/500=325/500=0.65.

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3) P(A)=0.40, P(B)=0.35, P(C)=0.30, P(A∩B)=0.18, P(A∩C)=0.12, P(B∩C)=0.11, P(A∩B∩C)=0.05. Find P(A∪B∪C).

Explanation

0.40+0.35+0.30−0.18−0.12−0.11+0.05=0.69.

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4) |A|=100, |B|=90, |C|=60, |A∩B|=40, |A∩C|=30, |B∩C]=20, |A∩B∩C|=0, total n=200. Find P(A∪B∪C).

Explanation

(100+90+60−40−30−20+0)/200=160/200=0.80.

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5) P(A)=0.50, P(B)=0.40, P(A∪B)=0.72. Find P(A∩B).

Explanation

P(A∩B)=P(A)+P(B)−P(A∪B)=0.50+0.40−0.72=0.18.

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6) |A|=120, |B|=100, |C|=80, |A∩B|=50, |A∩C|=40, |B∩C|=35, total n=200, and |A∪B∪C|=180. Find |A∩B∩C|.

Explanation

120+100+80−(50+40+35)+x=180 ⇒ 300−125+x=180 ⇒ x=5.

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7) P(A)=0.50, P(B)=0.40, P(A∩B)=0.20. Find P(exactly one of A or B).

Explanation

P(A only)+P(B only)=P(A)+P(B)−2P(A∩B)=0.50+0.40−0.40=0.50.

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8) P(A)=0.40, P(B)=0.35, P(C)=0.30, P(A∩B)=0.18, P(A∩C)=0.12, P(B∩C)=0.11, P(A∩B∩C)=0.02. Find P(none of A, B, or C).

Explanation

First P(union)=0.40+0.35+0.30−0.18−0.12−0.11+0.02=0.66, so P(none)=1−0.66=0.34.

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9) P(A)=0.50, P(B)=0.35, P(C)=0.25, P(A∩B)=0.20, P(A∩C)=0.15, P(A∩B∩C)=0.08. Find P(A only).

Explanation

P(A only)=P(A)−P(A∩B)−P(A∩C)+P(A∩B∩C)=0.50−0.20−0.15+0.08=0.23.

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10) P(A∩B)=0.18, P(A∩C)=0.14, P(B∩C)=0.12, P(A∩B∩C)=0.05. Find P(exactly two of A, B, C).

Explanation

Exactly two events = (0.18 + 0.14 + 0.12) – 3(0.05) = 0.29.

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Given P(A)=0.58, P(B)=0.46, P(A∩B)=0.31, find P(A∪B).
|A|=240, |B|=180, |A∩B|=95, total n=500. Find P(A∪B).
P(A)=0.40, P(B)=0.35, P(C)=0.30, P(A∩B)=0.18, P(A∩C)=0.12,...
|A|=100, |B|=90, |C|=60, |A∩B|=40, |A∩C|=30, |B∩C]=20,...
P(A)=0.50, P(B)=0.40, P(A∪B)=0.72. Find P(A∩B).
|A|=120, |B|=100, |C|=80, |A∩B|=50, |A∩C|=40, |B∩C|=35,...
P(A)=0.50, P(B)=0.40, P(A∩B)=0.20. Find P(exactly one of A or B).
P(A)=0.40, P(B)=0.35, P(C)=0.30, P(A∩B)=0.18, P(A∩C)=0.12,...
P(A)=0.50, P(B)=0.35, P(C)=0.25, P(A∩B)=0.20, P(A∩C)=0.15,...
P(A∩B)=0.18, P(A∩C)=0.14, P(B∩C)=0.12,...
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