A high school class conducts a survey where students are asked about their eye color and whether or not they wear glasses. The two-way table below shows the results of the survey as relative frequencies.
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A. Based on the results of the survey, what is the probability, rounded to the nearest tenth, that a student had blue eyes and wore glasses?
B. Based on the results of the survey, what is the probability, rounded to the nearest tenth, that a student's eye color is brown given that the student wears glasses?
Word Bank:
16% 4% 24% 8% 12% 20% 42.9%
Blank 1:
Blank 2:
A. Ah, the famous blue-eyed glasses-wearers! Let's see, according to the table, the relative frequency of students with blue eyes and wearing glasses is 8%. So, the probability, rounded to the nearest tenth, is 8%.
B. Now, this one's a bit tricky. We need to find the probability that a student's eye color is brown given that they wear glasses. Looking at the table, we see that out of all the glasses-wearing students, 20% have brown eyes. So, the probability, rounded to the nearest tenth, is 20%. Good luck finding those stylish brown-eyed glasses wearers!
A. Based on the results of the survey, the probability that a student had blue eyes and wore glasses can be calculated by finding the relative frequency in the cell where "Blue" and "Glasses" intersect. From the two-way table, this intersection has a relative frequency of 0.02 (or 2%). Rounding this to the nearest tenth, the probability is 2%. Therefore, the answer for Blank 1 is "2%".
B. Based on the results of the survey, the probability that a student's eye color is brown given that the student wears glasses can be calculated by finding the conditional relative frequency. The conditional relative frequency is calculated by dividing the relative frequency of the cell where "Brown" and "Glasses" intersect by the sum of the relative frequencies of all cells where "Glasses" is marked. From the two-way table, the relative frequency of the intersection of "Brown" and "Glasses" is 0.08 (or 8%), and the sum of the relative frequencies of all cells where "Glasses" is marked is 0.14 (or 14%). Therefore, the conditional relative frequency is 0.08 / 0.14 = 0.5714 (or 57.14%). Rounding this to the nearest tenth, the probability is 57.1%. Therefore, the answer for Blank 2 is "57.1%".
To find the probability that a student had blue eyes and wore glasses, we need to look at the intersection of those two categories in the two-way table.
A. To find the probability that a student had blue eyes and wore glasses, we locate the cell that corresponds to the intersection of the "Blue" row and the "Wears Glasses" column in the table. From the table, we see that the relative frequency in that cell is 8%. Therefore, the probability, rounded to the nearest tenth, is 8%.
To find the probability that a student's eye color is brown given that the student wears glasses, we need to look at the conditional probability.
B. To find this probability, we focus on the column "Wears Glasses" and look at the relative frequencies of each eye color within that column. From the table, we see that the relative frequency of "Brown" within the "Wears Glasses" column is 12%. Therefore, the probability, rounded to the nearest tenth, is 12%.
Therefore, the answers are:
A. The probability that a student had blue eyes and wore glasses is 8%. (Blank 1: 8%)
B. The probability that a student's eye color is brown given that the student wears glasses is 12%. (Blank 2: 12%)
A. The relative frequency for students with both blue eyes and glasses is 0.08 or 8%. Therefore, the probability, rounded to the nearest tenth, that a student had blue eyes and wore glasses is 8%.
B. The relative frequency for students who wear glasses given that they have brown eyes is 0.2 or 20%. The relative frequency for students who wear glasses in general is 0.24 or 24%. Therefore, the conditional probability that a student's eye color is brown given that the student wears glasses is:
(0.2) / (0.24) = 0.8333...
Rounded to the nearest tenth, the probability is 42.9%.