Express the probability 0.0000000121 in scientific notation, as a percentage, and as odds of one in N.

Step-by-step solution

1. 1

   Move the decimal point to make a mantissa between 1 and 10. Starting from $0.0000000121$, sliding the point right past the eight leading zeros lands on $1.21$. Each rightward step divides by $10$, so eight steps must be compensated by $10^{-8}$:

   $0.0000000121=1.21\times 10^{-8}$

   The requirement $1\le |{\rm mantissa}|<10$ is what fixes the exponent uniquely — $12.1\times 10^{-9}$ is the same number but not in standard form.

2. 2

   Check the conversion by expanding.

   $1.21\times 10^{-8}=\frac{1.21}{100{,}000{,}000}=0.0000000121$

   Counting the zeros confirms eight decimal places before the $121$.

3. 3

   Convert to a percentage. A percentage is the number multiplied by $100\%$, which shifts the point two places right (or raises the exponent by $2$):

   $1.21\times 10^{-8}\times 100\%=1.21\times 10^{-6}\%=0.00000121\%$

   The percent sign is not decoration here: writing $1.21\times 10^{-8}\%$ would be a hundredfold error.

4. 4

   Express it as odds. "One chance in $N$" means $N=1/p$:

   $N=\frac{1}{1.21\times 10^{-8}}=82{,}644{,}628$

   so the event is about 1 in 82.6 million — a far more intuitive statement of the same figure.

5. 5

   Note the significant figures. Only the digits $1$, $2$, $1$ carry information; the eight leading zeros are placeholders. So the value has three significant figures, and rounding it to $1.2\times 10^{-8}$ (about 1 in 83 million) is the two-figure version.