If you work in a Bayesian framework, updating to take account
of additional information can be easy. [Note: Bayesians tend to use
terminology probability interval or credible interval instead
of confidence interval.]
Prior distribution. Take the probability $\theta$ of brushing as a beta random variable
with values in $(0,1).$ Without much information about a fresh subject, if you feel their probability $\theta$ of brushing has $P(1/3 <\theta < 2/3) = 0.60,$ you may decide on a prior distribution $\mathsf{Beta}(3,3),$ with density
$$f(\theta)\propto \theta^{3-1}(1-\theta)^{3-1},$$
where the symbol $\propto$ (pronounced 'proportional to') indicates that, for simplicity, we have omitted the constant factor that makes $f(\theta)$ integrate to unity over $(0,1).$ If you want to start
with a 95% probability interval that matches this prior distribution,
then it would be $(.15,\, .85).$
qbeta(c(.2, .8), 3,3)
[1] 0.3265979 0.6734021
qbeta(c(.025,.975), 3,3)
[1] 0.1466328 0.8533672
[Instead of guessing you might use an interval based on behavior of people you consider to be somehow 'similar'.]
Likelihood information. Each brushing or non-brushing is a Bernoulli trial with
$1$ for brushing and $0$ for not. Maybe over the first two days,
this person brushes 3 out of the four times of interest. Then
their binomial likelihood is
$$ g(x|\theta) \propto \theta^3(1-\theta).$$
Posterior distribution. Then by Bayes theorem the updated interval estimate is based on the
posterior distribution $h(\theta|x)$ found by multiplying the prior distribution by the likelihood, as follows:
$$h(\theta|x) = f(\theta)\times g(x|\theta) \\
\propto \theta^{3-1}(1-\theta)^{3-1} \times \theta^3(1-\theta)\\
\propto \theta^{6-1}(1-\theta)^{4-1},$$
which we recognize as the 'kernel' (density without norming constant)
of the distribution $\mathsf{Beta}(6,4),$ so that the new
95% interval estimate is $(.30,\,.86).$
qbeta(c(.025, .975), 6,4)
[1] 0.2992951 0.8630043
Subsequent updating. Then this posterior distribution becomes the prior distribution
for the next update. Each updated distribution adds $1$ to the first beta shape parameter for a brushing or adds $1$ to the second
shape parameter for a non-brushing. For example, if the next
event is a brushing, then the next updated 95% interval is
changed slightly to $(.35,\, .88).$
qbeta(c(.025,.975), 7,4)
[1] 0.3475471 0.8784477
Moving window updates. If you continue increasing the beta
shape parameters over a long period of time, then the updated 95% probability intervals will get ever narrower. At some point, you may feel
that behavior in the distant past may no longer be of current interest. Then you could base updates on some recent window of time (say, the last
three months), dropping earlier counts as later ones become available.
Someone with a three-month history of brushing $3/4$ of the time (on your twice daily schedule)
would have an updated 95% probability interval of about $(.65,.83).$
qbeta(c(.025,.975), 70, 24)
[1] 0.6523714 0.8271333
Notes: (1) If you need point estimates at each step, you can use the mean of of the current beta posterior distribution. The distribution $\mathsf{Beta}(\alpha,\beta)$ has mean $\mu = \frac{\alpha}{\alpha+\beta}.$
(2) One Bayesian probability interval, the Jeffreys Interval,
is often used as a frequentist confidence interval. Based on a non-informative prior distribution, it uses appropriate
quantiles of $\mathsf{Beta}(x + .5, n-x + .5)$ when there are $x$ successes in $n$ trials. You can see the Wikipedia
article on 'binomial confidence intervals' for more about this interval estimate.
(3) Depending on your interests, you may want to Google topics such as 'sequential Bayesian updating'.
