What you did in the OP is exactly correct. Derive a decimal just how you did, chances of event occuring/all possibilities and then do what you did:
Divide 1 by the decimal. That gives you the price a theoretical fair bookmaker would give you. The odds are that minus 1, to 1.
So
tossing a coin, chance of heads is .5
Price offered 1/.5 = $2
Odds (2-1) = 1 to 1 or even money
tossing a dice, chance of a 6 is .1666666
1/.166666 = $6
Odds (6-1) = 5 to 1
If you want to know what the odds in a billion are, first you take the decimal you got by dividing 34 by 30.3 billion, and divide it by 30.3 billion again. Now you have your chance per billion in decimal form:
0.0000000001122
To convert this to percentage, multiply by 100:
0.00000001122% in a billion chance of getting killed in an encounter with a dog.
That’s going to be hard to say in words, as you’ve discovered, so you probably want to get it closer to one. In order to do that, you are going to need to multiply BOTH your billion and your percent odds until the percent odds is as close to one as it’s going to get. You have seven zeros after the decimal up there, so you need seven zeros in the number you’re multiplying it BY to get a whole number:
0.00000001122
x 10,000,000
= 1.22%
1,000,000,000
x 10,000,000
= 10,000,000,000,000,000
Your odds of getting killed by a dog in any given single encounter in the United States are about one in ten quadrillion. If you want to get even closer to one, you can do additional multiplication and division until you’re as close as your heart desires.
Ugh, I screwed up on the zeroes. Let’s try again…
If you want to know what the odds in a billion are, first you take the decimal you got by dividing 34 by 30.3 billion, and divide it by 30.3 billion again. Now you have your chance per billion in decimal form:
0.00000000001122
To convert this to percentage, multiply by 100:
0.000000001122% in a billion chance of getting killed in an encounter with a dog.
That’s going to be hard to say in words, as you’ve discovered, so you probably want to get it closer to one. In order to do that, you are going to need to multiply BOTH your billion and your percent odds until the percent odds is as close to one as it’s going to get. You have eight zeros after the decimal up there, so you need eight zeros in the number you’re multiplying it BY to get a whole number:
0.000000001122
x 100,000,000
= 1.122%
1,000,000,000
x 100,000,000
= 100,000,000,000,000,000
Your odds of getting killed by a dog in any given single encounter in the United States are about one in a quintillion.
UGH I can’t number today. That’s not a quintillion, it’s 400 quadrillion.
Serves me right trying to answer a math question the day after a Christmas party…
Either this (and all that follows) is completely wrong, or I’ve completely misunderstood what you are trying to do.
If you want to express the probability in terms of “… in a billion”, you would multiply the probability by one billion.
Why would you divide it by 30.3 billion twice?
Because the first time you’re finding the ratio of deaths to encounters, and the second time you’re taking BOTH that number AND the 30.3 billion and reducing it to odds per ONE billion.
The problem is really about converting the relationship between the two numbers into a form that is relatively easy to express verbally. You can change the numbers to anything you want as long as they maintain the same ratio.
It’s completely possible that I am either doin’ it wrong, or expressing myself poorly. I probably shouldn’t even have tried, I went to a holiday party last night and had too much champagne so my brain isn’t braining too well.
I’m running with “doin’ it wrong” here, but I hope it was a good party. 
It was a very good party! And now that I’m not hung over, it’s embarrassing how wrong I was doin’ it. :smack: You would never guess that I was a straight A math student, LOL!