Problem 8 · The urn with blue balls
Probability with replacement and a quadratic equation.
Integer answer, at most 4 digitsAn urn contains one black ball, four white balls and some blue balls (more than one). A ball is drawn at random, its colour is noted and it is returned to the urn. Then another ball is drawn at random. For what number of blue balls is the probability that the two drawn balls have the same colour equal to $1/2$?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: with replacement, the two draws are independent: $P(\text{same colour}) = \sum P(\text{colour})^{2}$.
With $b$ blue balls there are $b + 5$ balls in total:
$$P = \frac{1^{2} + 4^{2} + b^{2}}{(b+5)^{2}} = \frac{b^{2} + 17}{(b+5)^{2}} = \frac{1}{2}.$$
Cross-multiply and simplify:
$$2b^{2} + 34 = b^{2} + 10b + 25 \;\Longrightarrow\; b^{2} - 10b + 9 = 0 \;\Longrightarrow\; b = 1 \ \text{or}\ b = 9.$$
Since the statement says there is more than one blue ball, $b = 9$.
Answer: 9