Numeracy Skills

2.5. Probability

The islanders of Baal, still concerned by the rate at which the talking-cat population is expanding, have decided, against the wishes of the zoologists, to cull a significant proportion of these in order to stem this increase.

The cats, however, hearing of this plan, have other ideas. One of the brightest of these puts forward a suggestion. If all (talking) cats remain silent, he argues, it will be impossible for the islanders to tell apart talking-cats from their mute relatives and this being so, the Baalanders, who are known for their great love of this native creature, will be forced to call off their campaign.

Realising this duplicitous action, the Baalanders have to make a choice. They reason that, since the vast majority of felines will be talking-cats, if they cull a sizeable percentage of the overall cat population - say, 20,000 - then the chances are that only a few non-talkers will be destroyed and this is perhaps a price worth paying.

Nevertheless, if they were to select a single cat entirely at random, they would like to know what the chances or probability would be of its being a native non-talker.

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To draw an analogy here, if you were to select an individual card from a deck, assuming it to be a standard 52 card pack (Jokers removed), the chances of selecting, for example, the 7 of clubs (♣) would be 1/52. Thus the probability of selecting any particular card at random would be 1/52 (or, for every 52 attempts, one should be successful). The probability of selecting any spade would be 1/4, since there are only four suits in a pack. The chance of picking either a red or a black card would be 1/2 and the chances of picking out a numbered or head card (including aces), for similar reasons, would be 36/52 and 16/52 respectively. It is also possible to work out the probability of picking out a succession of cards. The chances of picking out two red cards in a row, for example, would be 1/2 x 1/2 (1/4); the chances of picking out a red card, then a club would be 1/2 x 1/4 (1/8) and so on.

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Equations such as these are particularly helpful when it comes to risk calculation - as we see with the situation that confronts the islanders of Baal, not to mention their cats!

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Optional Activity

Have a go at a few easy probability questions on this Mathopolis site to help improve your probability skills before moving on. This is an optional activity, but well worth a go.

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Applying the above reasoning to the Baalanders case, if there were an equal proportion of talking and non-talking cats (50:50), then the probability of selecting either cat at random would be 1/2. Given that the ratio is 32,000:500, however, the equation is slightly trickier. See if you can work it out!

Since the overall cat population is 32,500 and the number of talking-cats, 32,000, the probability would be 32,000/32,500. But this is rather a large number to handle! We could break it down further:

32,000/32,500 = 16,000/16,250 = 8,000/8,125 etc.,

but in this case, it might be easier to convert such a value to a percentage.

Going back to our earlier method of doing this, we would be left with the following equation:

(32,000 / 32,500) x 100 = 98.46.

The probability of selecting a talking cat at random, then, would be 98.46%.

Fortunately, there is a clause written into the Baalanders’ constitution which states that, for any activity which might endanger a native feline, unless this has a greater than 99% safety rate, it must not be pursued. So, for now at least, the cats are safe...!