QuestionA survey of 50 students shows 25 like online classes and 35 are freshmen. Find counts for events $A$ , $B$ , $A \cap B$ , and neither.

Studdy Solution

STEP 1

Assumptions1. Total number of students surveyed is50. . Event A represents students who like to take online classes.
3. Event B represents first year students.
4. The probability of a student liking online classes given they are a freshman is $\frac{15}{35}$ .
5. The probability of a student being a freshman given they like online classes is $\frac{15}{25}$ .

STEP 2

First, we need to find the number of students who like online classes and are also first year students. This is represented by the intersection of events A and B, denoted as $A \cap B$ . We can find this by using either of the conditional probabilities given.Let's use $(A \mid B)=\frac{15}{35}$ , which means the probability of a student liking online classes given they are a freshman.We know that $(A \mid B) = \frac{A \cap B}{B}$ , so we can solve for $A \cap B$ .
$A \cap B =(A \mid B) \times B$

STEP 3

Now, plug in the given values for $(A \mid B)$ and B to calculate $A \cap B$ .
$A \cap B = \frac{15}{35} \times35$

STEP 4

Calculate the number of students who like online classes and are also first year students.
$A \cap B = \frac{15}{35} \times35 =15$

STEP 5

Now we can find the number of students who like online classes but are not first year students. This is represented by $A - A \cap B$ . $A - A \cap B = A -15$

STEP 6

Plug in the given value for A to calculate $A - A \cap B$ .
$A - A \cap B =25 -15$

STEP 7

Calculate the number of students who like online classes but are not first year students.
$A - A \cap B =25 -15 =10$

STEP 8

Similarly, we can find the number of first year students who do not like online classes. This is represented by $B - A \cap B$ . $B - A \cap B = B -15$

STEP 9

Plug in the given value for B to calculate $B - A \cap B$ .
$B - A \cap B =35 -15$

STEP 10

Calculate the number of first year students who do not like online classes.
$B - A \cap B =35 -15 =20$

STEP 11

Finally, we can find the number of students who neither like online classes nor are first year students. This is represented by the total number of students minus the sum of A and B.
$Total - (A + B) =50 - (25 +35)$

STEP 12

Calculate the number of students who neither like online classes nor are first year students.
$Total - (A + B) =50 - (25 +35) = -10$ However, this result is not possible as we cannot have a negative number of students. This discrepancy arises due to the fact that we have double counted the students who both like online classes and are first year students. So, we need to add back the number of students who fall into both categories.
$Total - (A + B) + A \cap B = -10 +15$

STEP 13

Calculate the number of students who neither like online classes nor are first year students.
$Total - (A + B) + A \cap B = -10 +15 =5$ So, the Venn diagram should be filled as follows-10 students like online classes but are not first year students. -20 students are first year students but do not like online classes. -15 students both like online classes and are first year students. -5 students neither like online classes nor are first year students.

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