In a previous post, we posed the following exercise to revise concepts from a yet earlier post

In the opening track of Jeff Wayne’s Musical version of [H.G. Wells’s] The War of the Worlds, we hear:

“The chances of anything coming from Mars, are a million to one,” he [Ogilvy the Astronomer] said

(AaaaaaaaaaH    aaaaaaah)

“The chances of anything coming from Mars, are a million to one,”

but still, they come!

Question: He said “chances”. Discuss.

There are a number of things going on here. By saying “a million to one” Ogilvy is invoking a ‘bookmakers odds’ way of talking. Perhaps HG Wells was a betting man, but I wouldn’t want to stake too much on that! Probably, it was intended to reflect how most people speak about probabilities – informally, and influenced more by references to betting than formal statistics.

The word ‘chance’ is usually meant to mean plain probability. Also note that betting odds are not necessarily an indication of actual probability per se – but the result of a bookmaker (or someone else) offering a risk/reward ratio that is consistent with their overall aims – and this involves considering how much money/pride/satisfaction is (possibly already) riding on various outcomes. If we are meant to interpret the remarks as Ogilvy’s scientific assessment – it would be cleaner if he expressed them for instance as

The risk/probability/chance (singular) of anything coming from Mars is 0.000001

This would be a rational way of reporting it. If we quibble, we may notice that just as the probability of “ten to one” is not 1/10 (0.1) but 1/11 (0.090909…), so the ‘probability’ exactly corresponding to odds of ‘a million to one’ is not precisely 0.000001, but 0.00000099999900000100 … 

Of course these differences mean nothing – especially as we are not reporting numbers which anyone has any right to know anyway – they are guesses.

Perhaps Ogilvy meant to say, as one would among men of learning in London at the time,

“I’d give million to one odds to anyone betting on anything coming from Mars”

Which, in the common tongue, would more likely be rendered as

“I’d give million to one odds against anything coming from Mars”

In fact, according to HG Wells (as opposed to Jeff Wayne) Ogilvy actually said

‘The chances against anything manlike on Mars are a million to one’

Note “chances of” vs “chances against”. As just noted, one might offer odds against something – but one does not normally talk of estimating probabilities ‘against’ something. If one said that – one would presumably mean the probability of something NOT happening, in which case one might be more formally consistent with the normal nomenclature of probability by saying:

‘The chances against anything manlike on Mars are 0.999999.’

‘The chances of anything manlike on Mars are 0.000001’

Or, losing the battle to retain the lyrical value of “million to ooone”:

‘The chances of anything manlike on Mars are one in a million’

But even estimating that we’re talking millions as opposed to billions or thousands is just a way of saying

‘The chances of anything manlike on Mars are very very small’

But let’s wait and see, ….