Let the probabilities of the 8 disjoint regions in the Venn diagram be:

We are given:

$P(\text{exactly one of }A\text{ or }B\text{ occurs})=\frac14.$

This means either $A$ occurs and $B$ does not, or $B$ occurs and $A$ does not. So,

$P(A\oplus B)=x_1+x_3+x_5+x_2+x_6+x_3=x_1+x_2+x_5+x_6+x_3?$

Instead of region-counting loosely, let us do it carefully:

• $A$ occurs and $B$ does not: regions $x_1, x_6$
• $B$ occurs and $A$ does not: regions $x_2, x_5$

Hence,

$x_1+x_2+x_5+x_6=\frac14 \qquad ...(1)$

Similarly,

$P(B\oplus C)=x_2+x_3+x_4+x_1?$

Carefully:

• $B$ occurs and $C$ does not: regions $x_2,x_4$
• $C$ occurs and $B$ does not: regions $x_3,x_6$

So,

$x_2+x_3+x_4+x_6=\frac14 \qquad ...(2)$

And,

• $C$ occurs and $A$ does not: regions $x_3,x_5$
• $A$ occurs and $C$ does not: regions $x_1,x_4$

Thus,

$x_1+x_3+x_4+x_5=\frac14 \qquad ...(3)$

We need:

$P(A\cup B\cup C)=x_1+x_2+x_3+x_4+x_5+x_6+x_7.$

Since $x_7=\frac{1}{16}$, it remains to find

$S=x_1+x_2+x_3+x_4+x_5+x_6.$

Now add (1), (2), and (3):

Grouping terms:

So,

Therefore,

$P(A\cup B\cup C)=S+x_7=\frac38+\frac1{16}=\frac6{16}+\frac1{16}=\frac7{16}.$

Checking options

• A: $\frac{7}{16}$ ✔
• B: $\frac{7}{64}$ ✘
• C: $\frac{3}{16}$ ✘
• D: $\frac{7}{32}$ ✘

Hence the correct answer is Option A.