Question

the table summarizes the distribution of ages and assigned group for 120 participants in a study. one of these participants will be selected at random. what is the probability of selecting a participant from group a, given that the participant is at least 10 years of age?
a. \\(\frac{5}{24}\\)
b. \\(\frac{5}{16}\\)
c. \\(\frac{2}{5}\\)
d. \\(\frac{5}{8}\\)
(table:

• 0-9 years: group a=15, group b=5, group c=20, total=40
• 10-19 years: group a=16, group b=16, group c=15, total=40
• +20 years: group a=9, group b=26, group c=5, total=40
• total: group a=40, group b=40, group c=40, total=120)

Explanation:

Step1: Identify Group A and "at least 10 years"

Group A total: 40. "At least 10 years" for Group A: 10 - 19 (16) + 20+ (9) = 25.

Step2: Calculate conditional probability

Conditional probability formula: $P(\text{at least 10} | \text{Group A}) = \frac{\text{Group A, at least 10}}{\text{Total Group A}} = \frac{16 + 9}{40} = \frac{25}{40} = \frac{5}{8}$? Wait, no, wait. Wait the table: Group A 0 - 9:15, 10 - 19:16, 20+:9. So "at least 10" is 16 + 9 = 25. Total Group A: 15 + 16 + 9 = 40. Wait but the options: d is 5/8? Wait no, wait the options: a. 5/24, b.5/16, c.2/5, d.5/8? Wait no, maybe I misread. Wait Group A: 0 - 9:15, 10 - 19:16, 20+:9. So "at least 10" is 16 + 9 = 25. Total in Group A: 15 + 16 + 9 = 40. So 25/40 = 5/8? But let's check again. Wait the problem says "given that the participant is at least 10 years of age"? No, wait: "given that the participant is from group A, given that the participant is at least 10 years of age"? Wait no, the question: "probability of selecting a participant from group A, given that the participant is at least 10 years of age". Wait, no: "selecting a participant from group A, given that the participant is at least 10 years of age". So numerator: Group A and at least 10: 16 + 9 = 25. Denominator: total at least 10: 40 (10 - 19 total) + 40 (20+ total)? Wait no, the table: 0 - 9 total:40, 10 - 19 total:40, 20+ total:40. So at least 10: 40 + 40 = 80? Wait no, 10 - 19:40, 20+:40, so total at least 10: 80. Group A in at least 10: 16 (10 - 19) + 9 (20+) = 25. So probability is 25/80? Wait no, that can't be. Wait I think I misread the table. Let's re-express the table:

Rows: 0 - 9, 10 - 19, 20+; Columns: Group A, Group B, Group C, Total.

0 - 9:
Group A:15, Group B:5, Group C:20, Total:40

10 - 19:
Group A:16, Group B:16, Group C:15, Total:40

20+:
Group A:9, Group B:26, Group C:5, Total:40

Total:
Group A:15+16+9=40, Group B:5+16+26=47? Wait no, the total row says Group A:40, Group B:40, Group C:40, Total:120. So my initial table reading was wrong. Let's correct:

0 - 9:
Group A:15, Group B:5, Group C:20, Total:40 (15+5+20=40)

10 - 19:
Group A:16, Group B:16, Group C:8? Wait no, total 10 - 19 is 40, so 16 (A) + 16 (B) + C = 40 → C=8? But the original table has Group C 15? No, the user's table: "10-19 Years" row, Group A:16, Group B:16, Group C:15, Total:40? 16+16+15=47≠40. So there's a typo, but the total row says Group A:40, so 15 (0-9) + 16 (10-19) + 9 (20+) = 40. So 20+ Group A:9, 10-19:16, 0-9:15. So "at least 10" for Group A:16 + 9 = 25. Total participants at least 10: total 120 - 40 (0-9) = 80. Wait, no: the question is "given that the participant is at least 10 years of age", so we are restricted to the "at least 10" group (size 80). In that group, how many are from Group A? 16 (10-19) + 9 (20+) = 25. So probability is 25/80 = 5/16? Wait 25 divided by 80: divide numerator and denominator by 5: 5/16. Ah, that's option b.

Wait, let's redo:

Step1: Find total number of participants at least 10 years old: total - 0-9 total = 120 - 40 = 80.

Step2: Find number of participants in Group A and at least 10 years old: Group A 10-19 (16) + Group A 20+ (9) = 25.

Step3: Conditional probability = (Group A and ≥10) / (Total ≥10) = 25/80 = 5/16.

Yes, that matches option b.

Answer:

b. $\frac{5}{16}$