There are 11 pens in a box, 8 are black, 3 are red. Two pens are taken out at random without replacement. What is the probability the pens are the same colour?

For the pens to be the same colour either two black pens need to be selected OR two red pens. So start with the probability of two black pens BB: On the first selection the probability of taking a black pen is 8/11 (the number of black pens in the box divided by the total number of pens in the box). On the second selection the probability of taking a black pen AGAIN is 7/10 (the number of black pens remaining in the box divided by the total number of pens remaining , which is 10 as remember they are taken without replacement). This means the probability of taking two black pens without replacement is 8/11 X 7/10 = 56/110 (multiply the two probabilities together). Now we have to work out the probability of two red pens RR: For the first selection the probability of taking a red pen is 3/11 (number of red pens in the box divided by total number of pens in box). For the second selection the probability of taking a red pen AGAIN is 2/10 (number of red pens remaining in box divided by total remaining number of pens). This means the probability of taking two red pens without replacement is 3/11 X 2/10 = 6/110 (again multiply the two probabilities together). So the probability of BB is 56/110 and RR is 6/110. To calculate the total probability of the two pens being the same colour we add these two together 56/110 + 6/110 = 62/110. (You add the two probabilities at the end as in the original statement you need both pens to be black OR both pens to be red and OR always means you add those probabilities together)