QuestionFind the number of debt holders with student loans (Event $A$ ) in a sample where 13 have student loans and 30 have credit card debt, given P(A|B) = $\frac{11}{30}$ .

Studdy Solution

STEP 1

Assumptions1. The probability of a debt holder having student loan debt, given that they also have credit card debt, is $\frac{11}{30}$ . . There are13 people with student loan debt (Event A).
3. There are30 people with credit card debt (Event B).

STEP 2

The probability of a debt holder having student loan debt, given that they also have credit card debt, is the number of people who have both student loan and credit card debt divided by the total number of people with credit card debt.In mathematical terms, this probability is represented as $(A|B) = \frac{(A \cap B)}{(B)}$ , where $(A|B)$ is the probability of event A given event B, $(A \cap B)$ is the probability of both events A and B occurring, and $(B)$ is the probability of event B.

STEP 3

We know that $(A|B) = \frac{11}{30}$ , and we want to find $(A \cap B)$ , which is the number of people who have both student loan and credit card debt.We can rearrange the formula from step2 to solve for $(A \cap B)$ $(A \cap B) =(A|B) \times(B)$

STEP 4

We know that $(B)$ , the total number of people with credit card debt, is30.So, we can plug in the values for $(A|B)$ and $(B)$ to calculate $(A \cap B)$ $(A \cap B) = \frac{11}{30} \times30$

STEP 5

Calculate the number of people who have both student loan and credit card debt $(A \cap B) = \frac{11}{30} \times30 =11$

STEP 6

Now, we can fill in the Venn diagram.The number of people with both student loan and credit card debt is11, which is the intersection of events A and B.The number of people with only student loan debt is the total number of people with student loan debt minus the number of people with both types of debt.Similarly, the number of people with only credit card debt is the total number of people with credit card debt minus the number of people with both types of debt.

STEP 7

Calculate the number of people with only student loan debt $Number\, of\, people\, with\, only\, student\, loan\, debt = Total\, number\, of\, people\, with\, student\, loan\, debt - Number\, of\, people\, with\, both\, types\, of\, debt$ $Number\, of\, people\, with\, only\, student\, loan\, debt =13 -11 =2$

STEP 8

Calculate the number of people with only credit card debt $Number\, of\, people\, with\, only\, credit\, card\, debt = Total\, number\, of\, people\, with\, credit\, card\, debt - Number\, of\, people\, with\, both\, types\, of\, debt$ $Number\, of\, people\, with\, only\, credit\, card\, debt =30 -11 =19$ So, in the Venn diagram, the number of people with only student loan debt is2, the number of people with both types of debt is11, and the number of people with only credit card debt is19.

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