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Difficulty: Hard

	$0$ – $9$ years	$10$ – $19$ years	$20 plus$ years	Total
Group A	$5$	$17$	$8$	$30$
Group B	$6$	$8$	$16$	$30$
Group C	$19$	$5$	$6$	$30$
Total	$30$	$30$	$30$	$90$

The table summarizes the distribution of age and assigned group for $90$ 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?

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Explanation

Choice B is correct. Since the participant will be selected at random, the probability of selecting a participant from group A, given that the participant is at least $10$ years of age, is equal to the number of participants from group A who are at least $10$ years of age divided by the total number of participants who are at least $10$ years of age. Based on the table, in group A, there are $17$ participants who are $10$ – $19$ years of age and $8$ participants who are $20 plus$ years of age. Therefore, there are a total of $17 plus 8$ , or $25$ , participants in group A who are at least $10$ years of age. Based on the table, of the total number of participants, there are $30$ participants who are $10$ – $19$ years of age and $30$ participants who are $20 plus$ years of age. Therefore, a total of $30 plus 30$ , or $60$ , of the participants are at least $10$ years of age. Thus, the probability of selecting a participant from group A, given that the participant is at least $10$ years of age, is $StartFraction 25 Over 60 EndFraction$ , or $\frac{5}{12}$ .

Choice A is incorrect. This is the number of participants from group A who are at least $10$ years of age divided by the total number of participants, rather than divided by the number of participants who are at least $10$ years of age.

Choice C is incorrect. This is the probability of randomly selecting a participant from group A, given that the participant is $10$ – $19$ years of age, rather than given that the participant is at least $10$ years of age.

Choice D is incorrect. This is the probability of randomly selecting a participant who is at least $10$ years of age, given that the participant is in group A.