# Calculating probabilities from incomplete data

First of 2 part query: I have observed a 100 doctors and recorded the number of times they touch surfaces while in a room:

\begin{array}{l|c|c|c|c} \text{Dr number}&\text{Bed}&\text{Table}&\text{Chair}&\text{Door}\\ \text{Dr } 1&3&2&2&1\\ \text{Dr } 2&4&1&0&1\\ \vdots&\vdots &\vdots &\vdots &\vdots\\ \text{Dr } 100& 2&3&1&1 \end{array}

I would like to calculate the probability of him touching the Table for example. Would it be:

\begin{array}{l|c|c|c|c|c} &\text{Bed}&\text{Table}&\text{Chair}&\text{Door}&\text{Total}\\ \text{Average contacts}&3&2&1&1&\text{7}\\ P(\text{surface})&3/7&2/7&1/7&1/7&1 \end{array}

This is confusing me but I think a fresh pair of eyes will spot it instantly.

2nd part: I gave questionnaires out to some doctors who ticked boxes instead of giving contact numbers, i.e.:

\begin{array}{l|c|c|c|c} \text{Dr number}&\text{Bed}&\text{Table}&\text{Chair}&\text{Door}\\ \text{Dr} &\checkmark&\checkmark&\checkmark&\checkmark\\ \text{Dr } \checkmark&\checkmark&\checkmark&\times&\checkmark\\ \vdots&\vdots &\vdots &\vdots &\vdots\\ \text{Dr } 100& \checkmark&\checkmark&\checkmark&\checkmark \end{array}

What can I deduce from this table of ticks to compare with the table of exact values? Or is it not worth anything?

EDIT: Revised Surface contact Probability Should this replace table 2?

\begin{array}{l|c|c|c|c|c} &\text{Bed}&\text{Table}&\text{Chair}&\text{Door}&\text{Total}\\ \text{Total contacts}&9&6&3&3&\text{21}\\ P(\text{surface})&9/21&6/21&3/21&3/21&1 \end{array}

• Are you looking for the conditional expectation of how many tables they touched, given the total surfaces touched? Or are you not making any inferences from how much surfaces were touched? – Andrew May 5 '12 at 16:53
• The second part is conditional probability – Andrew May 5 '12 at 16:54
• Hi @Andrew let's say that there is only one of each surface in the room and I have recorded each time they touch it. So I'd like to deduce what the probability is of touching the table given all my 100 observations. BUT I would ALSO like to know what the expectation of touches is for each surfaces. On what would they be conditional? Could you clarify your second comment for me please? Cheers – HCAI May 5 '12 at 17:10
• Ah I thought that there could be more than one surface in each room, and that is how I framed my answer. Let me think about it a bit :p – Andrew May 5 '12 at 17:42
• If you have no additional information about the new doctor we are predicting for-(how many things he will touch, which rooms he will be in), then the probability should just be the proportion of the 100 who touched the table, and the expectation should be the mean. When I answered earlier for some reason I though you had additional information. – Andrew May 5 '12 at 17:45