Question

a group of 125 people were asked if they own a car and a bicycle. 35 said they own a bicycle. 66 said they own a car. 40 said they own neither. find the probability of owning a bicycle given you own a car. draw a venn diagram to model the scenario.

Explanation:

Step1: Find the number of people who own at least one (car or bicycle)

Total people \(n(T)=125\), people who own neither \(n(N) = 35\).
Number of people who own at least one \(n(A\cup B)=n(T)-n(N)=125 - 35=90\)

Step2: Use the formula \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\)

Let \(n(A)\) (number of car - owners) \(=40\), \(n(B)\) (number of bicycle - owners) \(=66\), \(n(A\cup B) = 90\)
Substitute into the formula: \(90=40 + 66-n(A\cap B)\)
\(n(A\cap B)=40 + 66-90=16\)

Step3: Find the probability \(P(B|A)\)

By the formula for conditional probability \(P(B|A)=\frac{n(A\cap B)}{n(A)}\)
Since \(n(A\cap B) = 16\) and \(n(A)=40\)
\(P(B|A)=\frac{16}{40}=\frac{2}{5}=0.4\)

Step4: Draw the Venn diagram

• Draw two overlapping circles labeled \(C\) (for car) and \(B\) (for bicycle) inside a rectangle (universal set).
• The overlapping region (intersection) has \(16\) (people who own both).
• The non - overlapping part of the car circle: \(40 - 16=24\)
• The non - overlapping part of the bicycle circle: \(66 - 16 = 50\)
• Outside the two circles (but inside the rectangle) has \(35\) (people who own neither)

Answer:

The probability that a person owns a bicycle given that they own a car is \(0.4\)