[tex]\begin{matrix}&\text{chocolate}&\text{vanilla}&\text{total}\\\text{children}&0.14&0.26&0.40\\\text{adults}&0.21&0.39&0.60\\\text{total}&0.35&0.65&1.00\end{matrix}[/tex]
a. "Chocolate" and "Adults" (whatever those mean) will be independent as long as
[tex]P(\text{chocolate}\cap\text{adults})=P(\text{chocolate})\cdot P(\text{adults})[/tex]
"Chocolate" has the marginal distribution given by the second column, with a total probability of [tex]P(\text{chocolate})=0.35[/tex]. Similarly, "Adults" has the marginal distribution described by the third row, so that [tex]P(\text{adults})=0.60[/tex]. Then
[tex]P(\text{chocolate})\cdot P(\text{adults})=0.35\cdot0.60=0.21[/tex]
Meanwhile, the joint probability of "Chocolate" and "Adults" is given by the cell in the corresponding row/column, with [tex]P(\text{chocolate}\cap\text{adults})=0.21[/tex].
The probabilities match, so these events are indeed independent.
Parts (b) and (c) are checked similarly.
b. Yes;
[tex]P(\text{children})\cdot P(\text{chocolate})=0.40\cdot0.35=0.14[/tex]
[tex]P(\text{children}\cap\text{chocolate})=0.14[/tex]
c. Yes;
[tex]P(\text{vanilla})\cdot P(\text{children})=0.65\cdot0.40=0.26[/tex]
[tex]P(\text{vanilla}\cap\text{children})=0.26[/tex]
