Question

The two tables below show the success rates of several groups of businesses in a certain city. The first shows the number of businesses of several types started in Sharon’s city over the course of two years, and the number of those businesses which did not succeed and were forced to shut down within two years of opening. The second deals with separate records of successful new businesses, showing how much profit those new businesses turned over two years. Businesses on the boundary lines fall in the lower category.

Type
Food
Retail
Financial
Service
Opened
3,193
2,280
1,898
5,045
Closed
1,977
1,626
1,443
3,548


Up to $25k
$25-50k
$50-75k
$75-100k
Over $100k
Food
945
623
601
258
114
Retail
813
548
347
188
63
Financial
316
244
195
86
51
Service
979
739
432
174
124

Using the tables as experimental data, determine which of the following situations have a probability of at least 15.00%.

I. a food establishment succeeding and earning $50,000 or more
II. a service establishment succeeding and earning between $25,000 and $75,000
III. a retail establishment succeeding and earning no more than $50,000
a.
I only
b.
I and II
c.
II and III
d.
III only

Answer 1

Interpret each event as: pick a business of the given type; the probability the event occurs = P(success | type) × P(profit range | success, type). Use the first table for P(success | type) and the second table for P(profit range | success, type).

Compute success counts:
- Food: 3193 − 1977 = 1216 → P(success | food) = 1216/3193 ≈ 0.38099
- Retail: 2280 − 1626 = 654 → P(success | retail) = 654/2280 ≈ 0.28684
- Service: 5045 − 3548 = 1497 → P(success | service) = 1497/5045 ≈ 0.29667

Compute profit proportions (from the second table):
- Food, ≥ $50k: (601+258+114)/2541 = 973/2541 ≈ 0.38292
- Service, $25–75k: (739+432)/2448 = 1171/2448 ≈ 0.47828
- Retail, ≤ $50k: (813+548)/1959 = 1361/1959 ≈ 0.69484

Now multiply:
I. Food succeed and ≥ $50k: 0.38099 × 0.38292 ≈ 0.14595 = 14.60% (< 15%)
II. Service succeed and $25–75k: 0.29667 × 0.47828 ≈ 0.14192 = 14.19% (< 15%)
III. Retail succeed and ≤ $50k: 0.28684 × 0.69484 ≈ 0.19925 = 19.93% (≥ 15%)

Only III reaches at least 15.00%. Answer: d.