Stage 9 - Q1 - 2 marks
\(P(A)=\frac{3}{14}\). Find \(P(\text{not }A)\).
Hint
The probabilities of an event and its complement sum to one.
Stage 9 - Q2 - 2 marks
A bag has 6 red and 3 blue counters. Draw twice, replacing and mixing after the first draw. Are the two red events independent?
Hint
Compare the second-draw probabilities after each possible first colour.
Stage 9 - Q3 - 2 marks
A and B are independent. \(P(A)=\frac{2}{3}\), \(P(B)=\frac{2}{5}\). Find \(P(\text{A and not B})\).
Hint
Translate the requested outcome into one probability for each event.
Stage 9 - Q4 - 3 marks
A spinner lands on gold 11 times in 110 spins. Use these results to predict gold in 220 future spins. Is the prediction guaranteed?
Hint
The observed fraction is an estimate for future trials.
Stage 9 - Q5 - 3 marks
Use the independent-event tree to find \(P(\text{exactly one of A and B})\).
Hint
Multiply along a path; add only the separate paths you need.
Stage 9 - Q6 - 3 marks
Design a random-digit simulation for red, blue and green with probabilities \(\frac{3}{10}\), \(\frac{2}{10}\) and \(\frac{5}{10}\). Use digits 0–9.
Hint
Each uniform random digit has probability one tenth.
Stage 9 - Q7 - 3 marks
Choose uniformly from 1 to 10. A means even and B means at most 7. Test independence.
Hint
Count B separately among even and odd outcomes.
Stage 9 - Q8 - 2 marks
A and B are independent. \(P(A)=\frac{1}{6}\), \(P(B)=\frac{3}{4}\). Find \(P(\text{neither A nor B})\).
Hint
Translate the requested outcome into one probability for each event.
Stage 9 - Q9 - 3 marks
A spinner lands on gold 48 times in 60 spins. Use these results to predict gold in 180 future spins. Is the prediction guaranteed?
Hint
The observed fraction is an estimate for future trials.
Stage 9 - Q10 - 6 marks
Two independent spinner stages each succeed with probability \(\frac{2}{3}\). A prize is won for exactly one success. In 180 plays, 85 prizes occur. Find the theoretical probability, predicted count and observed relative frequency; explain the difference.
Hint
Add the two one-success paths before comparing experiment and theory.