Question

1. you surveyed people about their dessert preference.

(there is a venn diagram with two circles labeled ice cream and cake. in the ice cream only part is 31, in the intersection is 23, in the cake only part is 17, and outside both circles is 9. also, 31 + 23 + 17 + 9 = 80 is written with total next to it.)
a) $p(\text{cake} \cap \text{ice cream})$ (with a hand - written solution: $\frac{23}{80} \approx 0.23$)
b) are the preferences of cake and ice cream independent? (with some hand - written calculations and the answer no, theyre not independent)
c) $p(\text{cake} \cup \text{ice cream})$ (with a hand - written solution: $\frac{31 + 23 + 17}{80}=\frac{71}{80} \approx 0.89$)
d) are the preferences of cake and ice cream mutually exclusive? (with a hand - written answer: no, blc some people chose both at the same time (where blc is likely because))
e) $p(\text{cake but not ice cream})$
f) $p(\text{neither cake nor ice cream})$

Explanation:

Step1: Confirm total survey size

Total = $31 + 23 + 17 + 9 = 80$

Step2: Solve part a) (intersection probability)

Identify overlap count, divide by total.
$P(\text{cake} \cap \text{ice cream}) = \frac{23}{80} = 0.2875$

Step3: Solve part b) (independence check)

First calculate individual probabilities:
$P(\text{cake}) = \frac{23 + 17}{80} = \frac{40}{80} = 0.5$
$P(\text{ice cream}) = \frac{31 + 23}{80} = \frac{54}{80} = 0.675$
Check if $P(\text{cake}) \times P(\text{ice cream}) = P(\text{cake} \cap \text{ice cream})$:
$0.5 \times 0.675 = 0.3375
eq 0.2875$

Step4: Solve part c) (union probability)

Sum all in either set, divide by total.
$P(\text{cake} \cup \text{ice cream}) = \frac{31 + 23 + 17}{80} = \frac{71}{80} = 0.8875$

Step5: Solve part d) (mutual exclusivity check)

Mutually exclusive means no overlap. The overlap count is 23, not 0.

Step6: Solve part e) (cake only probability)

Identify cake-only count, divide by total.
$P(\text{cake but not ice cream}) = \frac{17}{80} = 0.2125$

Step7: Solve part f) (neither probability)

Identify neither count, divide by total.
$P(\text{neither cake nor ice cream}) = \frac{9}{80} = 0.1125$

Answer:

a) $\boldsymbol{0.2875}$
b) No, the preferences are not independent, since $P(\text{cake}) \times P(\text{ice cream})
eq P(\text{cake} \cap \text{ice cream})$.
c) $\boldsymbol{0.8875}$
d) No, the preferences are not mutually exclusive, because 23 people prefer both desserts.
e) $\boldsymbol{0.2125}$
f) $\boldsymbol{0.1125}$