On a randomly selected school day, find the probability that Samantha goes to school by bus.

Given that Samantha went to school by bus on Monday, find the probability that it was raining.

In a randomly chosen school week, find the probability that Samantha goes to school by bus on exactly three days.

After \(n\) school days, the probability that Samantha goes to school by bus at least once is greater than \(0.95\). Find the smallest value of \(n\).

appropriate approach     (M1)

eg     \({\text{P}}(R \cap B) + {\text{P}}(R' \cap B)\), tree diagram,

one correct multiplication     (A1)

eg     \(0.2 \times 0.5,{\text{ }}0.24\)

correct working     (A1)

eg     \(0.2 \times 0.5 + 0.8 \times 0.3,{\text{ }}0.1 + 0.24\)

\({\text{P(bus)}} = 0.34 {\text{(exact)}}\)     A1     N3

[4 marks]

recognizing conditional probability     (R1)

eg     \({\text{P}}(A|B) = \frac{{{\text{P}}(A \cap B)}}{{{\text{P}}(B)}}\)

correct working     A1

eg     \(\frac{{0.2 \times 0.5}}{{0.34}}\)

\({\text{P}}(R|B) = \frac{5}{{17}},{\text{ }}0.294\)     A1     N2

[3 marks]

recognizing binomial probability     (R1)

eg     \(X \sim {\text{B}}(n,{\text{ }}p)\), \(\left( \begin{array}{c}5\\3\end{array} \right)\) \({(0.34)^3},{\text{ }}{(0.34)^3}{(1 - 0.34)^2}\)

\({\text{P}}(X = 3) = 0.171\)     A1     N2

[2 marks]

METHOD 1

evidence of using complement (seen anywhere)     (M1)

eg     \(1 - {\text{P (none), }}1 - 0.95\)

valid approach     (M1)

eg     \(1 - {\text{P (none)}} > 0.95,{\text{ P (none)}} < 0.05,{\text{ }}1 - {\text{P (none)}} = 0.95\)

correct inequality (accept equation)     A1

eg     \(1 - {(0.66)^n} > 0.95,{\text{ }}{(0.66)^n} = 0.05\)

\(n > 7.209{\text{   (accept }}n = 7.209{\text{)}}\)     (A1)

\(n = 8\)     A1     N3

METHOD 2

valid approach using guess and check/trial and error     (M1)

eg     finding \({\text{P}}(X \geqslant 1)\) for various values of n

seeing the “cross over” values for the probabilities     A1A1

\(n = 7,{\text{ P}}(X \geqslant 1) = 0.9454,{\text{ }}n = 8,{\text{ P}}(X \geqslant 1) = 0.939\)

recognising \(0.9639 > 0.95\)     (R1)

\(n = 8\)     A1     N3

[5 marks]