Question

the table summarizes the distribution of age and assigned group for 90 participants in a study.

	0 - 9 years	10 - 19 years	20+years	total
group b	4	5	21	30
group c	11	14	5	30
total	30	30	30	90

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 $\frac{1}{6}$
b $\frac{1}{4}$
c $\frac{11}{30}$
d $\frac{1}{2}$

Explanation:

Step1: Calculate number of participants at least 10 years old

We add the number of participants in the 10 - 19 years and 20+ years age - groups. The total number of participants in these two age - groups is \(30 + 30=60\).

Step2: Calculate number of participants from Group A who are at least 10 years old

We add the number of participants in Group A in the 10 - 19 years and 20+ years age - groups. That is \(11 + 4 = 15\).

Step3: Use conditional - probability formula

The formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). In the case of counting problems, if \(A\) is the event of selecting a participant from Group A and \(B\) is the event of selecting a participant at least 10 years old, then \(P(A|B)=\frac{n(A\cap B)}{n(B)}\), where \(n(A\cap B)\) is the number of elements in the intersection of \(A\) and \(B\) and \(n(B)\) is the number of elements in \(B\). So the probability of selecting a participant from Group A given that the participant is at least 10 years old is \(\frac{15}{60}=\frac{1}{4}\).

Answer:

B. \(\frac{1}{4}\)