QuestionFind the numbers for a Venn diagram where $P(A|B) = \frac{8}{14}$ , with 32 using transit and 14 owning cars.

Studdy Solution

STEP 1

Assumptions1. The probability that a person uses public transit given that they also own a car is $\frac{8}{14}$ . . There are32 people who use public transit.
3. There are14 people who own cars.
4. Event $A$ represents the people who use public transit.
5. Event $B$ represents the people who own cars.

STEP 2

The probability that a person uses public transit given that they also own a car is $\frac{8}{14}$ . This is a conditional probability, represented as $(A|B)$ . $(A|B) = \frac{8}{14}$

STEP 3

In terms of the number of people, this probability means that out of every14 people who own a car,8 of them use public transit.

STEP 4

Since there are14 people who own cars in our sample, we can directly apply this ratio to find the number of car owners who also use public transit.
$Number\, of\, car\, owners\, who\, use\, public\, transit =14 \times \frac{8}{14}$

STEP 5

Calculate the number of car owners who also use public transit.
$Number\, of\, car\, owners\, who\, use\, public\, transit =14 \times \frac{8}{14} =8$

STEP 6

Now, we know that8 people who own cars also use public transit. This is the intersection of events $A$ and $B$ , represented as $A \cap B$ .

STEP 7

The number of people who use public transit but do not own a car can be found by subtracting the number of people who both own a car and use public transit from the total number of people who use public transit.
$Number\, of\, people\, who\, use\, public\, transit\, but\, do\, not\, own\, a\, car = Total\, number\, of\, people\, who\, use\, public\, transit - Number\, of\, car\, owners\, who\, use\, public\, transit$

STEP 8

Plug in the values to calculate the number of people who use public transit but do not own a car.
$Number\, of\, people\, who\, use\, public\, transit\, but\, do\, not\, own\, a\, car =32 -8$

STEP 9

Calculate the number of people who use public transit but do not own a car.
$Number\, of\, people\, who\, use\, public\, transit\, but\, do\, not\, own\, a\, car =32 -8 =24$

STEP 10

The number of people who own a car but do not use public transit can be found by subtracting the number of people who both own a car and use public transit from the total number of people who own a car.
$Number\, of\, people\, who\, own\, a\, car\, but\, do\, not\, use\, public\, transit = Total\, number\, of\, people\, who\, own\, a\, car - Number\, of\, car\, owners\, who\, use\, public\, transit$

STEP 11

Plug in the values to calculate the number of people who own a car but do not use public transit.
$Number\, of\, people\, who\, own\, a\, car\, but\, do\, not\, use\, public\, transit =14 -8$

STEP 12

Calculate the number of people who own a car but do not use public transit.
$Number\, of\, people\, who\, own\, a\, car\, but\, do\, not\, use\, public\, transit =14 -8 =6$ In the Venn diagram, the number of people who use public transit but do not own a car is24 (Event $A$ only), the number of people who own a car but do not use public transit is6 (Event $B$ only), and the number of people who both own a car and use public transit is8 (Intersection of Events $A$ and $B$ ).

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