There are only red counters, yellow counters and blue counters in a bag.

Kevin takes at random a counter from the bag. He puts the counter back in the bag. Lethna takes at random a counter from the bag. She puts the counter back in the bag.

The probability that both counters are red or that both counters are yellow is 13/36.

The probability that the first counter is red and the second counter is not red is 1/4.

Seb takes at random a counter from the bag.

Work out the probability that Seb takes a yellow counter.


So far, from the text, I've formed $2$ equations but since the number of unknowns is greater than the number of equations, I cannot solve.

$[(R^2)/(R+Y+B)^2] + [(Y^2)/(R+Y+B)^2] = 13/36$

$[(R)/(R+Y+B)]*[(Y+B)/(R+Y+B)] = 1/4$

Any help would be appreciated. Thanks.

  • $\begingroup$ I expect it's easier to work with probabilities. So, let $R, Y,C$ be the respective probabilities. Then $R+B+C=1$ is automatic, so you can eliminate one variable right at the start. $\endgroup$ – lulu Feb 7 at 15:16
  • $\begingroup$ Got it, probability of obtaining a yellow is 1/3 i think. Thanks for the advice. $\endgroup$ – Ahmed Hussain-Shah Feb 7 at 15:35
  • $\begingroup$ Inspired guesswork can also give an answer, recognizing that $13 = 3^2 + 2^2$ and $36 = 6^2.$ $\endgroup$ – David K Feb 7 at 15:41

Let $R$, $Y$ and $B$ be the events in which a red, yellow and blue counter is drawn from the bag, respectively. We have:

$$P(R) \cdot (1 - P(R)) = \frac{1}{4} \iff P(R)^2 - P(R) + \frac{1}{4} = 0 \iff P(R) = \frac{1 \pm \sqrt{1 - 1}}{2} = \frac{1}{2}$$

$$P(R) \cdot P(R) + P(Y) \cdot P(Y) = \frac{13}{36} \iff P(Y)^2 = \frac{13 - 9}{36} = \frac{4}{36} = \frac{1}{9} \iff P(Y) = \frac{1}{3}$$

The probability that Seb draws a yellow counter, thus equals $\frac{1}{3}$.

  • $\begingroup$ Thank you, confirmed my answer. $\endgroup$ – Ahmed Hussain-Shah Feb 7 at 15:36
  • $\begingroup$ @I've upvoted your answer but can't see any option to tag the question as answered or close the question, I'm not very computer literate it seems. $\endgroup$ – Ahmed Hussain-Shah Feb 7 at 15:43
  • $\begingroup$ You accepted the answer, which means you as questioner think this a good answer for the question. You have probably not upvoted it, because it still has 0 reputation, that may be due to the fact that you yourself (just started) have not much reputation and certain rights only come with certain reputation. There is no need to close the answer, usually when people see that there is an accepted answer they know that is probably no longer necessary to look at the question "to help the questioner". $\endgroup$ – Ingix Feb 7 at 18:14

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