Question

note: please make sure to properly format your answers. all dollar - figures in the answers need to include the dollar sign and any amount over 1,000 should include the comma ($2,354.67). all percentage values in the answers need to include a percentage sign (%). for all items without specific rounding instructions, round your answers to two decimal places, show both decimal places (5.06). the following venn diagram describes cars sold during last summer at pennys autoland. the letter f represents four - wheel drive, s represents satellite radio, and r represents remote start. a. how many people bought cars with four - wheel drive? b. how many people bought cars with satellite radio and remote start? c. how many people purchased cars with all three options? d. how many people purchased cars with exactly one of these options? e. how many people purchased cars at pennys autoland last summer? f. a buyer from this group is selected at random to receive lifetime free oil changes. what is the probability that they purchased remote start? g. one person from the group that bought four - wheel drive is going to be selected at random. find p(f). h. determine if f and r are independent or associated events.

Explanation:

Step1: Find number of cars with four - wheel drive

Sum the numbers in the 'F' circle. $19 + 8+5 + 7=39$

Step2: Find number of cars with satellite radio and remote start

Sum the numbers in the intersection of 'S' and 'R'. $7+10 = 17$

Step3: Find number of cars with all three options

The number in the intersection of all three circles is 7.

Step4: Find number of cars with exactly one option

Sum the numbers in the non - overlapping parts of each circle. $19+16 + 20=55$

Step5: Find total number of cars sold

Sum all the numbers in the Venn diagram. $19+8 + 16+5+7+10+20=85$

Step6: Find probability of purchasing remote start

Number of cars with remote start is $5 + 7+10+20=42$. Probability $P(R)=\frac{42}{85}\approx0.49$

Step7: Find $P(F)$ for a car from the four - wheel drive group

Since we are already in the four - wheel drive group, $P(F) = 1$ (because we are selecting from the set of cars with four - wheel drive).

Step8: Check independence of F and R

$P(F)=\frac{39}{85}$, $P(R)=\frac{42}{85}$, $P(F\cap R)=\frac{5 + 7}{85}=\frac{12}{85}$. Since $P(F)\times P(R)=\frac{39}{85}\times\frac{42}{85}=\frac{1638}{7225}
eq\frac{12}{85}=P(F\cap R)$, F and R are associated events.

Answer:

a. 39
b. 17
c. 7
d. 55
e. 85
f. 0.49
g. 1
h. Associated events