QuestionFind the numbers for a Venn diagram where $P(A|B) = \frac{11}{30}$ , 45 own homes, and 30 live in San Francisco.

Studdy Solution

STEP 1

Assumptions1. The probability that a person owns a home, given that they live in San Francisco, California is $\frac{11}{30}$ . . Event $A$ represents owning a home.
3. Event $B$ represents living in San Francisco, California.
4. There are45 people in the sample who own homes.
5. There are30 people in the sample who live in San Francisco, California.

STEP 2

First, we need to find the number of people who both own a home and live in San Francisco, California. This is represented by the intersection of events $A$ and $B$ , denoted as $A \cap B$ . We can find this number by multiplying the total number of people who live in San Francisco, California by the given conditional probability.
$A \cap B = Total\, number\, of\, people\, in\, San\, Francisco \\times Probability\, of\, owning\, a\, home\, given\, living\, in\, San\, Francisco$

STEP 3

Now, plug in the given values for the total number of people in San Francisco and the conditional probability to calculate the number of people who both own a home and live in San Francisco.
$A \cap B =30 \\times \frac{11}{30}$

STEP 4

Calculate the number of people who both own a home and live in San Francisco.
$A \cap B =30 \\times \frac{11}{30} =11$

STEP 5

Now that we have the number of people who both own a home and live in San Francisco, we can find the number of people who own a home but do not live in San Francisco. This is represented by $A - A \cap B$ . We can find this number by subtracting the number of people who both own a home and live in San Francisco from the total number of people who own a home.
$A - A \cap B = Total\, number\, of\, home\, owners - Number\, of\, people\, who\, both\, own\, a\, home\, and\, live\, in\, San\, Francisco$

STEP 6

Plug in the values for the total number of home owners and the number of people who both own a home and live in San Francisco to calculate the number of people who own a home but do not live in San Francisco.
$A - A \cap B =45 -11$

STEP 7

Calculate the number of people who own a home but do not live in San Francisco.
$A - A \cap B =45 -11 =34$

STEP 8

Now that we have the number of people who own a home but do not live in San Francisco, we can find the number of people who live in San Francisco but do not own a home. This is represented by $B - A \cap B$ . We can find this number by subtracting the number of people who both own a home and live in San Francisco from the total number of people who live in San Francisco.
$B - A \cap B = Total\, number\, of\, people\, in\, San\, Francisco - Number\, of\, people\, who\, both\, own\, a\, home\, and\, live\, in\, San\, Francisco$

STEP 9

Plug in the values for the total number of people in San Francisco and the number of people who both own a home and live in San Francisco to calculate the number of people who live in San Francisco but do not own a home.
$B - A \cap B =30 -11$

STEP 10

Calculate the number of people who live in San Francisco but do not own a home.
$B - A \cap B =30 - =19$ The Venn diagram should be filled in as follows- The number of people who both own a home and live in San Francisco (the intersection of $A$ and $B$ ) is. - The number of people who own a home but do not live in San Francisco ( $A - A \cap B$ ) is34. - The number of people who live in San Francisco but do not own a home ( $B - A \cap B$ ) is19.

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