Hint
State independence before multiplying marginal probabilities. Multiply the two independent probabilities.
Worked solution
(a)
\(\displaystyle P(A\cap B)=P(A)P(B)=\frac{3}{20}\)
Answer: \(\displaystyle \frac{3}{20}\).
Answer: \(\displaystyle \frac{3}{20}\)
(b)
\(\displaystyle P(A)P(B^c)+P(A^c)P(B)=\frac{1}{2}\)
Answer: \(\displaystyle \frac{1}{2}\).
Answer: \(\displaystyle \frac{1}{2}\)
(c)
\(\displaystyle 1-(1-p)^r=\frac{83193}{100000}\)
Answer: \(\displaystyle \frac{83193}{100000}\).
Answer: \(\displaystyle \frac{83193}{100000}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Multiply the two independent probabilities.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not add probabilities for a sequence of required successes.
(b) · 2 marks
- 1 mark: Add the two disjoint ways to obtain exactly one signal.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: At least one includes cases with several successes.
(c) · 2 marks
- 1 mark: Use the complement of no successes.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not add probabilities for a sequence of required successes.
Total: 6 marks.