# Multiple binomial variables: probability they are all 1

We have observed four phone booths by recording, at a given time point, whether each phone booth is occupied (0 or 1). We have approximately 50 such observations (i.e. at 50 different time points, so each phone booth was recorded 50 times). Observations were pre-planned and spaced far enough apart so as to be independent -- the data were recorded over a period of about a month. How can I compute a confidence or credible interval for how often all four phone booths were occupied?

My thought process and ideas:

Observing each phone booth at a given time point is a Bernoulli trial. So we can model a phone booth as a binomial random variable with probability $p$ of being occupied. There are several ways to compute intervals for binomial variables (see this question). The trick is we want when they are all full.

Ideas:

1. Model each phone booth as a binomial. Using a $\text{Beta}(a, b)$ prior, the posterior is $\text{Beta}(y+a, n-y+a) = \text{Beta}(y+a, 50-y+a)$ where we observed $y$ times it was occupied. Then, draw posterior samples of all four phone booths separately and approximate $p(\text{all four are occupied})$ by counting the proportion of samples for which all four are occupied. Cons: assumes the four phone booths are independent, when really there is a such a thing as rush hour vs. down times, so there should be some correlation, perhaps even more relevant since we are interested in the "surge" periods when all are full.

2. Like the above, but as a multivariate normal so that a covariance matrix can be used so that observations are not assumed to be independent. Cons: doubtful that normal model is a good approximation for binomial data on small $N$.

3. At any given time, either 0, 1, 2, 3, or 4 phone booths is/are occupied. So model these as a multinomial distribution (either frequentist, or with Dirichlet prior). Cons: Not sure if this captures the ordered structure 0 < 1 < 2 < 3 < 4, or if it needs to. My understanding is that multinomial is intended for separate categories, like political candidates in an election and you want to see voting percentages of each based on a sample of voters; not sure it applies to this case.

• Is your assumption of independence between phone booths (i.e.the spacing argument is about distance between phone booths) or independence between trials (and the spacing argument is about being far apart in time)? – Glen_b Jul 23 '16 at 3:56
• Exactly how were the times chosen? Are they random or not? If they are not random, we have to be concerned about the possibility that phone booth occupancy and observation time are dependent. – whuber Jul 23 '16 at 12:55
• @Glen_b the spacing argument is about spacing in time, so that we are not "observing" say the phone booths 50 times in one hour (or to the extreme, 50 times in one minute, which is not indicative at all of overall usage, and we didn't have resources to observe for long stretches and use Poisson for the counts, that's why we have "instant" binary observations). The observations on the same day are typically hours-apart. – alexperrone Jul 23 '16 at 15:50
• @whuber, we have a planned protocol for the times so that each top-of-the-hour of the business day (8am-5pm) has an equal chance of being observed. Thus, the observations are counterbalanced by hour of the day (and also by day of the week). – alexperrone Jul 23 '16 at 15:53