Question

There are 5 red pens, 3 blue pens and 2 green pens in a box. Gary takes at random a pen from the box and gives the pen to his friend. He then takes at random another pen the box. What's the probability that both pens are the same colour?

Answer 1

To find the probability that both pens are the same color, we need to consider the different possible scenarios.

Scenario 1: Gary first takes a red pen and then another red pen.
The probability of the first event is 5/10 (5 red pens out of 10 total pens).
After the first event, there are 4 red pens and a total of 9 pens remaining, so the probability of the second event is 4/9.
Therefore, the probability of this scenario is (5/10) * (4/9) = 20/90.

Scenario 2: Gary first takes a blue pen and then another blue pen.
The probability of the first event is 3/10 (3 blue pens out of 10 total pens).
After the first event, there are 2 blue pens and a total of 9 pens remaining, so the probability of the second event is 2/9.
Therefore, the probability of this scenario is (3/10) * (2/9) = 6/90.

Scenario 3: Gary first takes a green pen and then another green pen.
The probability of the first event is 2/10 (2 green pens out of 10 total pens).
After the first event, there is 1 green pen and a total of 9 pens remaining, so the probability of the second event is 1/9.
Therefore, the probability of this scenario is (2/10) * (1/9) = 2/90.

Adding up the probabilities of the three scenarios, we get: (20/90) + (6/90) + (2/90) = 28/90.

Therefore, the probability that both pens are the same color is 28/90, which simplifies to 14/45.