Let's suppose that each of the 27.5M people has only 1 ticket to win a toy. If you are one of these people then the probability of not having a winning ticket is (27,500,000-1)/27,500,000 (almost 1). For this failure to occur 100 times (because there are 100 toys) is ((27,500,000-1)/27,500,000)100. So the probability of winning a toy is 1 less than this.
Probability of failure=(27499999/27500000)100.
We can use logarithms to compute this:
100log10(27499999/27500000)=-1.5793×10-6.
10-1.5793×10⁻⁶=0.9999963636 approx. (probability of failure)
Chances of winning 1-0.9999963636=0.0000036364 or about 1 in 275,000.5.
Note that this is almost the same as 100/27,500,000.
Another way of calculating this is to use the binomial expansion:
[(27500000-1)/27500000]100=(1-1/27500000)100=1-100/27500000 approx.
So 1-(1-100/27500000)=100/27500000=1/275000.