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Probability, 2ADV EQ-Bank 18

A survey of 50 students found that:

  • 28 students study Mathematics (set \(M\))
  • 22 students study Physics (set \(P\) )
  • 12 students study both Mathematics and Physics.
  1. How many students study Mathematics or Physics, or both?   (1 mark)

  2. If two students are chosen at random, what is the probability that both DO NOT study either Mathematics or Physics?   (2 marks)

Show Answers Only

a.
     

\(n(M \cup P)=38 \ \text {students}\)
 

b.    \(\dfrac{66}{1225}\)

Show Worked Solution

a.
     

\(n(M \cup P)=38 \ \text {students}\)
 

b.    \(P(M \cup P)=\dfrac{38}{50}\)

\(\text{Student 1:}\ P(\overline{M \cup P})=1-\dfrac{38}{50}=\dfrac{12}{50}\)

\(\text{Student 2:} \ P(\overline{M \cup P})=\dfrac{11}{49}\)

\(\therefore P\left(\text{Both study neither }\right)=\dfrac{12}{50} \times \dfrac{11}{49}=\dfrac{66}{1225}\)

Probability, 2ADV EQ-Bank 17

A survey of 85 households asked if they subscribed to the streaming services provided by Netflix (set \(N\)), Apple TV (set \(A\)), and Stan (set \(S\)).

The survey found that 17 households had no subscription and that \(n(N)=42, n(A)=35, n(S)=28\).

The survey also found

\(n(N \cap A)=18, n(N \cap S)=15, n(A \cap S)=12\)  and  \(n(N \cap A \cap S)=8\)

  1. Complete the Venn diagram below to accurately describe the information given.   (2 marks)


     

  2. A household is chosen randomly and is found to subscribe to Apple TV. What is the probability the household also subscribes to Stan?   (2 marks)

Show Answers Only

a.
           
 

b.    \(P(S \mid A)=\dfrac{12}{35}\)

Show Worked Solution

a.
           
 

b.    \(n(A)=35, n(A \cap S)=12\)

\(P(S \mid A)=\dfrac{n(A \cap S)}{n(A)}=\dfrac{12}{35}\)

Probability, 2ADV EQ-Bank 13

Consider the universal set  \(U=\{x\) is a positive integer and  \(x \leqslant 24\}\)

Three sets are defined as

\begin{aligned}
& A=\{x \text { is a factor of } 24\} \\
& B=\{x \text { is a perfect square}\} \\
& C=\{x \text { is divisible by } 3\}
\end{aligned}

  1. List the elements of set \(A\).   (1 mark)

  2. Find \(A \cap B\).   (1 mark)

  3. Find \((A \cup B) \cap C^c\)   (2 marks)

Show Answers Only

a.    \(A=\{1,2,3,4,6,8,12,24\}\)

b.    \(A \cap B=\{1,4\}\)

c.    \((A \cup B) \cap C^c=\{1,2,4,8,16\}\)

Show Worked Solution

a.    \(A=\{1,2,3,4,6,8,12,24\}\)
 

b.    \(B=\{1,4,9,16\}\)

\(A \cap B=\{1,4\}\)
 

c.    \(A \cup B=\{1,2,3,4,6,8,9,12,16,24\}\)

\(C=\{3,6,9,12,15,18,21,24\}\)

\(C^c=\{1,2,4,5,7,8,10,11,13,14,16,17,19,20,22,23\}\)

\((A \cup B) \cap C^c=\{1,2,4,8,16\}\)

Probability, 2ADV EQ-Bank 12

Consider the universal set  \(U=\{x\) is a positive integer and \(x \leqslant 15\}\)

Three sets are defined as:

\begin{aligned}
& A=\{x \text { is a multiple of } 3\} \\
& B=\{x \text{ is a prime number}\} \\
& C=\{x \text{ is even}\}
\end{aligned}

  1. List the elements of set \(A\).   (1 mark)

  2.  Find  \(B \cap C\)   (1 mark)

  3. Find  \(A \cup \overline{C}\)   (2 marks)

Show Answers Only

a.    \(A=\{3,6,9,12,15\}\)

b.   \(B \cap C = \{2\} \)

c.    \(A \cup \overline{C}=\{1,3,5,6,7,9,11,12,13,15\}\)

Show Worked Solution

a.    \(A=\{3,6,9,12,15\}\)
 

b.    \(B=\{2,3,5,7,11,13\}\)

\(C=\{2,4,6,8,10,12,14\}\)

\(B \cap C = \{2\} \)
 

c.    \(\text{Find} \ \ A \cup \overline{C}:\)

\(\overline{C}=\{1,3,5,7,9,11,13\}\)

\(A \cup \overline{C}=\{1,3,5,6,7,9,11,12,13,15\}\)

Probability, 2ADV EQ-Bank 11

Consider the universal set  \(U=\{1,2,3,4,5,6,7,8,9,10\}\).

Two sets, \(A\) and \(B\), are given as

\(A= \{1,3,4,7,9\}\)

\(B =\{2,4,7,10\}\)

  1. Find \(A \cup B\)   (1 mark)

  2. Find \(A \cap \overline{B}\)   (2 marks)

Show Answers Only

a.    \(A \cup B = \{1,2,3,4,7,9,10\}\)

b.    \(A \cap \overline{B} = \{3, 4, 9\}\)

Show Worked Solution

a.    \(A= \{1,3,4,7,9\},\ \ B=\{2,4,7,10\}\)

\(A \cup B = \{1,2,3,4,7,9,10\}\)
 

b.     \(\overline{B} = \{1,3,5,6,8,9\}\)

\(A \cap \overline{B} = \{3, 4, 9\}\)