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Problem-Solving and Data Analysis / Probability and conditional probability Difficulty: Hard

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

The table summarizes the distribution of age and assigned group for $90$ $90$ 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 $10$ $10$ years of age, is equal to the number of participants from group A who are at least $10$ $10$ years of age divided by the total number of participants who are at least $10$ $10$ years of age. Based on the table, in group A, there are $17$ $17$ participants who are $10$ $10$ – $19$ $19$ years of age and $8$ $8$ participants who are $20 +$ $20 plus$ years of age. Therefore, there are a total of $17 + 8$ $17 plus 8$ , or $25$ $25$ , participants in group A who are at least $10$ $10$ years of age. Based on the table, of the total number of participants, there are $30$ $30$ participants who are $10$ $10$ – $19$ $19$ years of age and $30$ $30$ participants who are $20 +$ $20 plus$ years of age. Therefore, a total of $30 + 30$ $30 plus 30$ , or $60$ $60$ , of the participants are at least $10$ $10$ years of age. Thus, the probability of selecting a participant from group A, given that the participant is at least $10$ $10$ years of age, is $\frac{25}{60}$ $StartFraction 25 Over 60 EndFraction$ , or $\frac{5}{12}$ $\frac{5}{12}$ .

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

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

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