# Probability to find a modern human with two teeth in different developmental stages

Each teeth growths from the crown to the root. There are different stages previously described to divide this process, as Crown initiation, one half of the crown, crown complete, root one quarter, root and a half, etc. Each of these stages of each different tooth (incisors, canines, molars, etc) has its own typical age of attainment (years) defined by the mean and the sd.

For example:

Incisor 1/4. This stage, defined by the acquisition of one quarter of the root, is attained in modern humans at a mean age of 5.28 years (SD=0.91)

Molar root complete. This stage, defined by the complete root lenght formation of the first molar, is attained in modern humans at a mean age of 8.45 years (SD=1.36).

You can see the normal distribution in the next figure:

Question 1: Which is the probability to find a modern human with these stages in both teeth? The answer must be in the overlapping area below both normal distributions, but I don't remember the procedure to solve it.

Question 2: Imagine we have one individual with an age of 5.5 years. Which is the probability that this individual at this age would have the incisor at that stage (1/4)? And which is the probability this individual would have first molar at stage of Root complete?

• Is this a homework problem? If so, it should have the self study' tag. – Peter Flom Jun 18 '14 at 11:01
• In fact it is not homework, is part of my research. If necessary, I can add that tag – antecessor Jun 18 '14 at 11:03
• OK. Then the first question is whether the growth of the two teeth is independent. I am not a dentist, but it seems very unlikely that the growth of different teeth in the same mouth is independent. If it is dependent, then you need to get (ideally) a bivariate distribution of growth; if you don't have that, then some other measure of the strength of relationship would help. – Peter Flom Jun 18 '14 at 11:14
• I am amazed. That seems completely impossible. You mean that there is no tendency for some children to get teeth earlier and others to get teeth later? – Peter Flom Jun 18 '14 at 11:20
• What does the event in question 1 mean? Doesn't this require age distribution (if most modern humans are adults, the probability of having entered both stages must be quite high?) – Juho Kokkala Jun 18 '14 at 14:14

OK, since the two events are independent, you can simulate this fairly easily (as long as you mean incisor 1/4 or more. If you want incisor 1/4 (and not yet 1/2) then you need the curve for incisor 1/2.

set.seed(123)

incisor14 <- rnorm(1000,5.28, 0.91)
molarcomp <- rnorm(1000,8.45,1.36)

#Question 1
sum(incisor14 > 5.28 & molarcomp > 8.45)/1000  #0.275

#Question 2a
sum(incisor14 < 5.5)/1000  #0.6
sum(molarcomp < 5.5)/1000 #0.013
`
• Can you clarify how you interpreted question 1 / how the simulation code relates to that? – Juho Kokkala Jun 18 '14 at 14:15
• I figured it was people who were above the mean on incisor and on molar. What's really needed is a table from previous data. But that isn't in the question. Other approaches might be better. – Peter Flom Jun 18 '14 at 14:34
• People have more than one molar and one incisor. How does your code account for that? (Notice the reference to "first" molar in question 2.) – whuber Jun 18 '14 at 15:00
• I just used the information in the post; but perhaps I misinterpreted it. The post didn't mention number of molars or incisors, so I assumed the data given was for the first molar and first incisor to grow and that the question was also about that. I think (but am not sure) that "first molar" refers to the earliest set of molars; we have 3 sets of molars, so that's 12 total The question could be clearer. – Peter Flom Jun 18 '14 at 15:22
• The comments in that post asked you to provide more details. You have not done so. The answer there was based on different guesses as to the details you left out. – Peter Flom Jun 18 '14 at 18:46