Each person in a group of students was identified by year and asked when he or she preferred taking classes: in the morning, afternoon, or evening. The results are shown in the table. Find the probability that the student preferred afternoon classes given he or she is a junior. Round your answer to the nearest thousandth.
Freshman Sophomore Junior Senior
Morning: 19 2 6 16
Afternoon: 17 3 13 15
Evening: 8 14 9 7
The total number of juniors is 28 (6+13+9). The number of juniors who preferred afternoon classes is 13. Therefore, the probability that a junior preferred afternoon classes is:
$$P(\text{Afternoon} \mid \text{Junior}) = \frac{13}{28} \approx \boxed{0.464}$$
To find the probability that a student preferred afternoon classes given that he or she is a junior, we need to calculate the following:
P(Afternoon | Junior) = P(Afternoon and Junior) / P(Junior)
Let's calculate the probability of a student preferring afternoon classes and being a junior (Afternoon and Junior):
P(Afternoon and Junior) = 13
Now, let's calculate the probability of a student being a junior (P(Junior)):
P(Junior) = Total number of junior students / Total number of students
Total number of junior students = 6 + 13 + 9 + 7 = 35
Total number of students = Sum of all the values in the table = 19 + 2 + 6 + 16 + 17 + 3 + 13 + 15 + 8 + 14 + 9 + 7 = 128
P(Junior) = 35 / 128
Now, let's substitute these values in the formula:
P(Afternoon | Junior) = 13 / (35 / 128)
P(Afternoon | Junior) ≈ 0.365
Therefore, the probability that a student preferred afternoon classes given he or she is a junior is approximately 0.365 or 36.5%.
To find the probability that a student preferred afternoon classes given that they are a junior, we need to divide the number of junior students who preferred afternoon classes by the total number of junior students.
From the table, we see that there are 13 junior students who preferred afternoon classes.
Next, we need to calculate the total number of junior students. To do this, we sum up the number of junior students in each preference category:
6 junior students preferred morning classes.
13 junior students preferred afternoon classes.
9 junior students preferred evening classes.
So, the total number of junior students is 6 + 13 + 9 = 28.
Now, we can calculate the probability by dividing the number of junior students who preferred afternoon classes by the total number of junior students:
Probability = Number of junior students who preferred afternoon classes / Total number of junior students = 13 / 28 ≈ 0.4643
Therefore, the probability that a student preferred afternoon classes given they are a junior is approximately 0.4643, rounded to the nearest thousandth.