It's been a while since I've done any probability problems, and I've created one for myself accidentally through a research and coding project. Please bear with me, its about death rates in the middle ages. Its a tad morbid, but let's sidestep that.
I found that only 50% of offspring made it to adulthood, with 30% not making it past toddlers. For my project I've decided that 5% of the population died at birth, 5% in infancy, 20% as toddlers, 15% as children, and 5% as teens.
I'm creating an emulator and would like to know how to calculate the chance of dying each 'year'. I've translated the years into 1 chance of death at birth, 3 chances in infancy, 6 in toddlerhood, 12 in childhood, and 8 in teenhood. (My scale is about 1 chance per 6 months actually.)
I also need to do this for the adults, but I'm hoping its the same formula and I'll be able to do those calculations myself after the first half is explained to me. In case I'm wrong about something, 10% of the population died as young adults and in my program they have 15 chances to fulfill this, 15% died as adults in 15 chances, 15% died as old people during 5 chances, and a lucky 10% lived out a full lifespan and died as elders, with 1 chance.
I've done some basic googling on probabilities, but I'm entirely unsure how to apply any of it to my problem. My problem seems to come from the fact that if a particular person didn't die yesterday, then what is the chance that they will live through today as well? I think this is conditional probability, but all the extra variables are just making it go right over my head. Any help is appreciated. Thanks!
I am not sure what you are looking for. For the purpose of this answer, I will interpret your question as the probability of dying in Stage $k$ (or equivalently surviving Stage $k$) given that one has made it into Stage $k$.
Start with the Stage Infancy. The proportion who make it to the beginning of infancy, that is, who survive Birth, is $0.95$. But $5\%$ die in infancy. So the probability of dying in infancy given one has made it there is $\frac{0.5}{0.95}\approx 0.0526316$. We give the numbers to absrd precision. Of course you should round off, perhaps to the nearest percent.
The pr0bability of surviving infancy given one has entered it is similarly $\frac{0.90}{0.95}\approx 0.9473684$. Note that this number is $1$ minus the $0.0526316$ we calculated in the previous paragraph. Since the conditional probability of surviving is so closely tied to the conditional probability of dying, from here on we only do one of the computations.
Now we find the probability of dying in toddlerhod, gven one has made it there. The probability we get to toddlerhood is $0.90$. The probability of not making it to the next stage is $0.20$. So the probability of dying as a toddler, given one has made it to that stage, is $\frac{0.20}{0.90}\approx 0.2222222$.
We have $70\%$ making it through toddler. But $15\%$ die as children, so the probability of dying as a child, given one has made it through toddler, is $\frac{0.15}{0.70}\approx 0.2142857$.
The probability of making it thogh the child stage is $0.55$. So the probability of dying as a teen, given we made it there, is $\frac{0.05}{0.55}\approx 0.0909091$.