Chance: Certain, Likely & Equally Likely

Chance questions on the test are really counting questions wearing different words. A box holds some coloured discs and one is taken without looking; then three statements are made and you say which are correct. Every word maps to a count: certain means every disc is that colour, impossible means none is, possible means at least one is, more likely means a bigger count, equally likely means the same count. So the first move is to write the count next to each colour. After that, each statement is a comparison you can settle in a second — five red against two green, eight blue against sixteen in total.

The statement that catches most students is 'equally likely to pick blue or not blue'. 'Not blue' is not a colour; it is everything else in the box added together. With 8 blue discs among 16, the other colours add to 8 as well, so the two outcomes are equally likely. The quick test: X or not X is equally likely exactly when X is half of the total. Two traps are common. One is comparing blue with the largest single other colour instead of with all of them. The other is treating 'not blue' as if it meant nothing, and calling the statement impossible.

With numbered counters the statements are about number properties: even or odd, greater than or less than, a particular number. Counting is still the whole method, but the boundaries need care. From 1 to 11 there are six odd numbers and only five even, because the list starts and ends on an odd number, so even and odd are not equally likely. 'Greater than 6' does not include 6, and neither does 'less than 6', so both groups have five counters and neither is more likely. And a single number is as likely as any other single number — the 9 is not harder to pick because it is bigger. Listing the qualifying numbers takes seconds and removes the guesswork.

A spinner turns the same idea into areas. What matters is the size of each colour's sector, not how many colours there are. On a spinner that is half red, a quarter blue and an eighth each of green and yellow, red or not red is equally likely, blue is twice as likely as green, and green and yellow are equally likely. When the sectors are equal, just count sectors per colour like discs in a box. Spinners also lead to 'about how many times' questions: with one yellow sector in eight, forty spins should give about five yellows. The word about is doing real work — chance says what to expect, not what will happen.

Harder items add a twist: something changes so that two outcomes become equally likely, and you must say what. For red against blue, the job is to close the gap between the two counts — add to the smaller colour or remove from the larger one, whichever the question allows. For red against not red, adding red discs grows one side only, so you keep adding until red equals everything else: five red against eleven others needs six more red, not three. Whatever the change, re-count both sides afterwards and check they match. Quiz and exercises now!