Grade 12 Counting & Probability Quiz 1 draws its questions at random from 34 questions across these topics:
A few of the question types, with full solutions, so you know what to expect.
The rules. For two events A and B, \(P(A)=0{,}4\), \(P(B)=0{,}5\) and \(P(A \text{ and } B)=0{,}2\). Determine \(P(A \text{ or } B)\).
Answer: \(0{,}7\)
Use the general addition rule, which is true for any two events:
\(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\)
\(= 0{,}4 + 0{,}5 - 0{,}2\)
Tip: The overlap is subtracted once, otherwise it is counted in both P(A) and P(B).
The rules. Events A and B are mutually exclusive, with \(P(A)=0{,}25\) and \(P(B)=0{,}35\). Determine \(P(A \text{ or } B)\).
Answer: \(0{,}60\)
Mutually exclusive means the events cannot both happen, so \(P(A \text{ and } B) = 0\).
\(P(A \text{ or } B) = 0{,}25 + 0{,}35 - 0\)
Tip: Only drop the subtraction when the question actually says mutually exclusive.
The rules. If \(P(A) = 0{,}72\), write down \(P(\text{not } A)\).
Answer: \(0{,}28\)
A and 'not A' are complementary: together they fill the whole sample space and cannot overlap.
\(P(\text{not } A) = 1 - P(A) = 1 - 0{,}72\)
The rules. A and B are independent events with \(P(A)=0{,}3\) and \(P(B)=0{,}6\). Determine \(P(A \text{ and } B)\).
Answer: \(0{,}18\)
For independent events the multiplication rule applies:
\(P(A \text{ and } B) = P(A) \times P(B) = 0{,}3 \times 0{,}6\)
Tip: Independent means MULTIPLY. Mutually exclusive means ADD. They are different ideas.
The rules. Given \(P(A)=0{,}6\), \(P(B)=0{,}3\) and \(P(A \text{ or } B)=0{,}72\). Are A and B independent?
Answer: Yes, because \(P(A \text{ and } B) = P(A)\times P(B) = 0{,}18\)
First find the intersection using the addition rule:
\(0{,}72 = 0{,}6 + 0{,}3 - P(A \text{ and } B)\)
\(\therefore P(A \text{ and } B) = 0{,}9 - 0{,}72 = 0{,}18\)
Now test independence: \(P(A)\times P(B) = 0{,}6 \times 0{,}3 = 0{,}18\)
Tip: To prove independence you must show both sides separately, then state that they are equal.