Hint
n is a fixed non-negative integer and \(\displaystyle 0\le p\le 1\). Choose the two successful trials and multiply the branch probabilities.
Worked solution
(a)
\(\displaystyle \binom{6}2p^2(1-p)^{4}=\frac{972}{3125}\)
Answer: \(\displaystyle \frac{972}{3125}\).
Answer: \(\displaystyle \frac{972}{3125}\)
(b)
\(\displaystyle 1-P(X=0)-P(X=1)=\frac{2396}{3125}\)
Answer: \(\displaystyle \frac{2396}{3125}\).
Answer: \(\displaystyle \frac{2396}{3125}\)
(c)
\(\displaystyle np=\frac{12}{5}\)
Answer: \(\displaystyle \frac{12}{5}\).
Answer: \(\displaystyle \frac{12}{5}\)
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: Choose the two successful trials and multiply the branch 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 confuse a binomial count with a geometric waiting time.
(b) · 2 marks
- 1 mark: Subtract the two excluded low counts.
- 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: Check strict and inclusive endpoints before using a cumulative probability.
(c) · 2 marks
- 1 mark: Use the binomial mean.
- 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 confuse a binomial count with a geometric waiting time.
Total: 6 marks.