Conditional probability and the Normal distribution – Year 13 A level maths test to download as PDF

80 students were asked whether they attend the gym.

1. Find the probability that a randomly chosen student attends the gym. [1 mark]
2. Find the probability that a student is in Year 13 given that they attend the gym. [1 mark]
3. Determine, with working, whether "attends the gym" and "Year 12" are independent. [2 marks]

Tom travels to school by bus with probability 0.6 and by bicycle otherwise. He is late with probability 0.1 when he takes the bus and 0.05 when he cycles.

1. Find the probability that Tom is late on a particular day. [2 marks]
2. Given that Tom is late, find the probability that he took the bus. [2 marks]
3. Find the probability that Tom is not late on any day of a 5-day week, assuming days are independent. [1 mark]

The mass of a tomato is modelled by \(X\sim N(64,9^2)\) grams.

1. Find \(P(X>70)\). [1 mark]
2. Find \(P(55\lt X<73)\). [1 mark]
3. Find the mass \(k\) such that 20% of tomatoes have mass less than \(k\). [2 marks]

\(X\sim N(\mu,\sigma^2)\) with \(P(X<20)=0.15\) and \(P(X>30)=0.4\).

1. Write down two equations connecting \(\mu\) and \(\sigma\). [2 marks]
2. Find \(\mu\) and \(\sigma\). [2 marks]

\(X\sim B(150,0.3)\). Use a Normal approximation with a continuity correction to estimate \(P(X\le40)\), and state the parameters of the approximating distribution.