There are 5 red balls and 7 green balls in a bag. A ball is taken from the bag at random and not replaced. Then a second ball is taken from the bag. What is the probability that the 2 balls are the same colour?

Initially, there are 12 balls in the bag. Therefore the chance of the first ball being red is 5/12 and green is 7/12. The probabilities of the second ball are dependent on the first ball, as it is removed from the bag: this is Conditional Probability.1st Case (Ball 1 is red):The number of red balls in the bag is now 4, and the total number of balls in the bag is 11. So the probability of pulling out a second red ball is 4/11. We aren't concerned with the probability of the green ball because we want to find the probability of both balls the same colour.2nd Case (Ball 1 is green):The number of green balls is now 6, total number of balls is 11. So probability of pulling out a green ball again is 6/11.
Now we have our two cases we need to find the probability of each and then the combined probability. For AND probability we multiply, so P(2 red) is (5/12) * (4/11) = 20/132 = 10/66P(2 green) is (7/12)*(6/11) = 42/132 = 21/66 Then we need to find the probability of either of the previous two outcomes occuring. Since this is OR probability, we add, therefore P(2 same colour) is 10/66 + 21/66 = 31/66.
Extra work could include identifying probability of both different without much more calculation (collective exhaustivity) or same again but balls are replaced after taking them from the bag.