How to combining individual probabilities of selecting something based on its attributes? Let's say that a box has a 50% chance of being selected based on it's colour and a 70% chance of being selected based on it's size. What is the overall probability of the box being selected?
Probability is not my very strong point but as far as I know, since it's a simple and question, you just multiply the probabilities, but then you get 0.5 * 0.7 which is 0.35. But then it seems wrong because according to my intuition the result should be more than 0.5. So either my intuition is wrong or that I don't know how to combine the probabilities.
 A: It is unanswerable without more information.  Suppose someone is totally colorblind... then the color of the box plays no impact on someone's decision making process on selecting the box or not.  It is not clear how our randomly selected person uses the information of color and size in their decision making process.
Now... it is possible that someone uses exactly one of the properties of size or color to influence their decision making process and which property is used is randomly selected with $p$ for the probability of using size to influence the decision and $(1-p)$ for using color.
We would have then a probability of choosing the right box as being $p\times 0.7 + (1-p)\times 0.5$
Of course, there is no reason to assume that someone uses only one attribute at a time to make a decision.  It is possible that knowledge of both the size and color will automatically allow the person to make the right decision every time, say for example we know that if it is a big golden box it is guaranteed to have a prize in it, but if we only went with "golden box" alone we might have grabbed small golden boxes which are empty or if we went with "big box" alone we might have grabbed big puke green boxes which are empty.
A: In this kind of question the answer depends on how you imagine the selecting takes place.
If you have, say, $100$ boxes, half red and half blue, and for each size box $35$ large and $15$ small then then if you pick a box at random it will be red with probability $50/100 = 1/2$ and large with probability $70/100 = 35/50 = 7/10$. It will be red and large with probability $0.7 \times 0.5 = 0.35$. That makes sense: there are $35$ of those boxes in the $100$. The product interpretation of "and" applies because the size and the color are independent.
But if all $50$ red boxes are large along with $20$ blue ones then a random box will be red and large with probability $1/2$.
There are other interpretations, suggested by @JMoravitz 's answer. Suppose there is just one red large box, which you are trying to select, while all the rest are small and blue, but your color perception is correct just half the time while your size perception is right $70\%$. Then you should just choose on the basis of size, winning with probability $0.7$.
So the probability of a large red box depends on context.
Can you imagine an experiment that leads to a probability greater than $0.7$?
