A jar contains blue and white beads in the ratio 3 : 1. Two beads are drawn together at random from the jar, and the probability that both beads are blue is \(\frac{11}{20}\). Find the total number of beads in the jar.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 16
Take the counts from the ratio as \(3k\) blue and \(k\) white. Drawing two beads is a selection, so P(both blue) \(=\) ways to choose 2 blue ÷ ways to choose any 2.
Total beads \(=4k\)
\(\displaystyle \text{P(both blue)}=\frac{{}^{3k}C_2}{{}^{4k}C_2}=\frac{3k(3k-1)}{4k(4k-1)}\)
\(\displaystyle \frac{3(3k-1)}{4(4k-1)}=\frac{11}{20}\)
\(60(3k-1)=44(4k-1)\)
\(180k-60=176k-44\), so \(4k=16\) and \(k=4\)
Total beads \(=4\times4=16\)