Question

question 3
3.1 a and b are two events such that:

• p(not a) = 0,55
• p(b) = 0,4
• p(a or b) = 0,67

are events a and b independent events? justify your answer with relevant calculations.
3.2 a survey was done among a group of 100 tourists to find out which city in south africa they loved during their visit. they chose from johannesburg (j), durban (d) and cape town (c). the results of the survey are listed below:

• 17 said they loved all three cities
• 30 said they loved durban and cape town
• 27 said they loved johannesburg and cape town
• 52 said they loved durban
• 9 said they loved only johannesburg and durban
• 12 said they loved johannesburg only
• 9 said they loved cape town only

the above information is represented in the partially completed venn diagram below:
3.2.1 write down the values of a, b, e and f.
3.2.2 calculate the probability that a tourist selected at random loved cape town or both johannesburg and durban.

Explanation:

Step1: Find P(A) for 3.1

We know that \(P(\text{not }A)=0.55\). By the formula \(P(A) = 1 - P(\text{not }A)\), so \(P(A)=1 - 0.55=0.45\).

Step2: Find \(P(A\cap B)\) for 3.1

Using the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Substitute \(P(A) = 0.45\), \(P(B)=0.4\) and \(P(A\cup B)=0.67\) into it. Then \(0.67=0.45 + 0.4-P(A\cap B)\), so \(P(A\cap B)=0.45 + 0.4-0.67 = 0.18\).

Step3: Check independence for 3.1

Two events \(A\) and \(B\) are independent if \(P(A\cap B)=P(A)\times P(B)\). Calculate \(P(A)\times P(B)=0.45\times0.4 = 0.18\). Since \(P(A\cap B)=P(A)\times P(B)\), events \(A\) and \(B\) are independent.

Step4: Find values for 3.2.1

For the V - enn diagram:

• The number of people who love all three cities is 17, so \(a = 17\).
• The number of people who love Durban and Cape - Town is 30, so \(b=30 - 17=13\).
• The number of people who love Johannesburg and Durban is 9, so \(e = 9\).
• The number of people who love Cape - Town only is 9, so \(f = 9\).

Step5: Calculate probability for 3.2.2

The number of people who love Cape - Town or both Johannesburg and Durban:
The number of people who love Cape - Town: \(17+13 + 9+27=66\). The number of people who love both Johannesburg and Durban is 9. But we have double - counted the 17 who love all three cities once.
The number of unique people who love Cape - Town or both Johannesburg and Durban is \(66+9 - 17=58\).
The probability \(P=\frac{58}{100}=0.58\).

Answer:

3.1: Events \(A\) and \(B\) are independent. Justification: \(P(A) = 0.45\), \(P(B)=0.4\), \(P(A\cap B)=0.18\) and \(P(A)\times P(B)=0.18\), so \(P(A\cap B)=P(A)\times P(B)\).
3.2.1: \(a = 17\), \(b = 13\), \(e = 9\), \(f = 9\)
3.2.2: \(0.58\)