A continuous random variable X takes values from 0 to 6 only and has a probability distribution that is symmetrical. Two values, a and b, of X are such that P(a < X < b) = p and P(b < X < 3) = 13/18 p, where p is a positive constant.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. [B1] Allow '=' in working but need an inequality in the answer. Allow 0<p<5/23
2. [M1] Any correct expression for the probability of a relevant region.
3. [A1]
4. [M1] Attempt to integrate with correct limits and equate to 5/27, oe. Integrate from 2 to 6 – a and equate 18/5 p = 2/3 Integrate from a to 3 and equate to 23/54 Integrate from 0 to a and equate to 2/27
5. [M1] For integrating and substitution of limits to form cubic in a.
6. [A1] Any correct three term cubic equation in a.
7. [M1] Attempt to factorise their cubic equation.
8. [A1] SC B1 for a = 1 only, if no method seen for solving the cubic.