SL 4.6—Venn diagrams

$P(T\cup B)=P(T)+P(W)-P(T\cap W)$, using the given counts out of 60.

$P(T)=35/60$, $P(W)=28/60$, $P(T\cap W)=15/60$.

$P(T\cup W)=\frac{35}{60}+\frac{28}{60}-\frac{15}{60}=\frac{48}{60}=0.80$.

80% of members use at least one of the two facilities.

$1.05$ comes from forgetting to subtract the intersection: $35/60+28/60=63/60$, an impossible probability greater than 1.

$0.25$ is just the intersection probability $15/60$, not the union.

$0.47$ incorrectly uses only the treadmill-only region divided by 60 rather than the full union.

Only $0.80$ correctly applies the addition rule with subtraction of the intersection.

$P(A\cup B)=P(A)+P(B)-P(A\cap B)=0.6+0.5-0.3=0.8$.

$P(A^{\prime }\cap B^{\prime })=1-P(A\cup B)=1-0.8=0.2$.

The probability neither event occurs is $0.20$.

$0.30$ is simply the intersection probability, not the complement of the union.

$0.10$ would result from mistakenly adding an extra subtraction of $P(A\cap B)$ again.

$0.80$ is the union itself, mistaken for its complement.

Only $0.20$ correctly represents $1-P(A\cup B)$.