Q.If two events A and B are such that $P(\bar A)=\dfrac{3}{10}$ and $P(A\cap\bar B)=\dfrac{1}{2}$ then $P(A\cap B)$ is:

Among 1 to 20, there are 6 multiples of 3, 5 multiples of 4, and 1 multiple of both (12); by inclusion-exclusion that's 10 numbers, giving probability 10/20 = 1/2.

Multiples of 3 in $\{1,\ldots,20\}$: 3,6,9,12,15,18 — 6 numbers.

Multiples of 4: 4,8,12,16,20 — 5 numbers.

Multiples of both 3 and 4 (i.e. of 12): 12 — 1 number.

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"Exactly one occurs" splits into two mutually exclusive cases: A but not B, or B but not A.

The event "exactly one of A, B occurs" is $(A\cap\bar B)\cup(\bar A\cap B)$. These two events are mutually exclusive (they can't both happen), so by the addition rule, …

$P(A\cap B)=\dfrac15$.

Since $P(\bar A)=\dfrac{3}{10}$, we get $P(A)=1-\dfrac{3}{10}=\dfrac{7}{10}$.

Event A splits into the disjoint parts $A\cap B$ and $A\cap\bar B$, so

$P(A)=P(A\cap B)+P(A\cap\bar B).$

Given $P(A\cap\bar B)=\dfrac12$: …

Mutually exclusive events cannot occur together, so $P(A\cap B)=0$ and the union rule simplifies to a plain sum.

For any two events: $P(A\cup B) = P(A)+P(B)-P(A\cap B)$.

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Since $A$ and $B$ share no outcomes, every outcome of $A$ automatically lies outside $B$, so $A\cap\bar B = A$.

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Since $A\cap B=\emptyset$, every outcome of $B$ lies outside $A$, so $\bar A\cap B = B$.

Mutually exclusive means $A\cap B=\emptyset$, so $B\subseteq\bar A$ (no outcome of $B$ is in $A$).

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'Exactly one occurs' splits into two disjoint cases whose probabilities simply add.

'Exactly one of $A$ and $B$ occurs' means either ($A$ occurs and $B$ does not) or ($B$ occurs and $A$ does not). In set notation these are $A\cap\bar B$ (A but not B) and $\bar A\cap B$ (B but not A). These two events are mutually exclusive (they cannot both happen at once), so by the addition rule for disjoint events: …

Count multiples of 3 (6 numbers), multiples of 4 (5 numbers), subtract the overlap — multiples of 12 (1 number) — to get 10 favourable outcomes out of 20, so the probability is $1/2$.

Multiples of 3 in $\{1,\ldots,20\}$: $3,6,9,12,15,18$ — that's $\lfloor 20/3\rfloor = 6$ numbers.

Multiples of 4: $4,8,12,16,20$ — that's $\lfloor 20/4\rfloor = 5$ numbers.

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