meaning of "a billion to one" and "50 million to 1"

Question

I'm reading Forensics: A Guide for Writers by D. P. Lyle. When the author talks about the probabilities of forensic testing results, he says:

1. 'No two people have the same DNA, but the testifying expert will not say that the DNA “absolutely matches” that of the defendant. Instead he will say that the probability that it matches is a billion to one. That is almost, but not quite, absolute.'

2. 'The probability that the theory holds true could be 50 percent (coin toss), 90 percent (probable), 99.999 percent (highly probable), or 50 million to 1 (virtually 100 percent probable).'

Does "a billion to one" mean "one in a billion" and "50 million to 1" mean "1 in 50 million"? I thought both phrases mean that something is extremely unlikely. But the context of the above two paragraphs seems to suggest that the two phrases mean that something is extremely likely.

Now I'm confused. Do they mean something is likely or unlikely? Can anybody explain the idea for me?

Answer 1

Does a "a billion to one" mean "one in a billion" and "50 million to 1" mean "1 in 50 million"

A billion to one is not the same as one in a billion. It isn't even the opposite. These are big numbers, though, so I am going to scale down a little.

Suppose you have ten red marbles and one blue marble in a drawer. You pull out 1 marble. The odds of that marble being red are 10 to 1. All the ways it could be red versus all the ways it couldn't.

The chance of that marble being red is 10 in 11 - all the ways it could be red versus all the ways it could be anything.

The likelihood of it being red is also 10 in 11. That's a "91 percent chance."

In the case of the DNA sample, a billion to one means that given the test results, there are very strong odds that the sample came from the defendant.

The likelihood of it coming from the defendant is 1000000000/1000000001.