A bag contains 64 coloured beads - Edexcel - A-Level Maths Statistics - Question 4 - 2018 - Paper 1

To show that r satisfies the equation, substitute the known values:

1. We know $y = 64 - r - 1$ then substituting gives: $r(64 - r - 1) = 240$. This expands to: [ r(63 - r) = 240 ] [ 63r - r^2 - 240 = 0 ] Reordering leads to: $r^2 - r - 240 = 0$.

Using the quadratic formula to solve for r:

1. From the equation $r^2 - r - 240 = 0$, use the quadratic formula: $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=1, b=-1, c=-240$.

2. Thus it becomes: $r = \frac{1 \pm \sqrt{1 + 960}}{2} = \frac{1 \pm 31}{2}$. Calculating gives: $r = 16 \text{ or } -15$. Therefore, the only valid solution is $r = 16$.

To compute this probability under the condition that at least one bead is red:

1. Calculate the total outcomes where at least one bead is red: $P(at least one red) = 1 - P(both not red)$ where: $P(both not red) = P(G) + P(Y) = \frac{(1)(y)}{64 \times 63}$. You need to also account for that the total combinations of 64 non-red beads are $\frac{y(y-1)}{64 \times 63}$.

2. Then calculate: $P(both red | at least one red) = \frac{P(both red)}{P(at least one red)}$ gives the required probability, simplifying should yield results near $\frac{16}{37}$.