# Notational ambiguity in probability

Let us consider the following question and answer

Question

A person can either take chemistry or physics, each with probability $\dfrac{1}{2}$. If takes chemistry, the probability of getting A grade is $\dfrac{1}{2}$. If takes physics, the probability of getting A grade is $\dfrac{1}{3}$. What is the probability that the person gets A in physics?

$$p(chemistry) = \dfrac{1}{2}$$ $$p(physics) = \dfrac{1}{2}$$ $$p(A/chemistry) = \dfrac{1}{2}$$ $$p(A/physics) = \dfrac{1}{3}$$ Now, $$p(A \cap physics) = p(physics) p(A/physics) = \dfrac{1}{2} \dfrac{1}{3}=\dfrac{1}{6}$$

The answer is correct and I can solve it mechanically. I used mechanically, because I don't know what is sample space in this problems and the events.

If we say that A is the event of getting A grade then what are the elements of the event? I am pretty confused. I can solve, but cannot write sample space and cannot justify events etc., please enlighten me on this.

Edit : Provide sample space, events in set form.

• The sample space consists of four possible outcomes: Chemistry with grade A, chemistry without grade A, physics with grade A and physics without grade A. – Arthur Jan 12 '18 at 12:45
• In this case the sample space can just be written as four elements based on the four events of interest. But you don't need to break everything down into an explicit sample space. Most of the time the sample space is just a mathematical abstraction without real world meaning. – Ian Jan 12 '18 at 12:51
• @Ian Is it okey to do problems intuitively without too much mathematical? because, i can able to solve, but feel abuse of notation – hanugm Jan 12 '18 at 12:57

The problem is phrased in a (slightly) confusing and incomplete way...perhaps that is the issue you have. As stated, you could not, for example, answer the question "what is the probability that the student gets at least one $A$" without making some assumptions. The problem is that we are not told whether taking Chemistry or Physics are mutually exclusive. As the question you are asked only concerns Physics, this ambiguity doesn't matter (and the answer you propose is good).

If you assume mutually exclusivity then the events are: $$(\text {Chemistry},\, A),\, (\text {Chemistry}, \text {not}\, A),\, (\text {Physics}, \,A),\, (\text {Physics, not}\, A)$$

If you do not assume mutual exclusivity then you have to allow for taking both or neither.

• Is Chemistry, A, not A, Physics are sets (Events)? – hanugm Jan 12 '18 at 12:53
• Each of those four are events. The first, say, means the student takes Chemistry and gets an $A$. – lulu Jan 12 '18 at 12:59
• Maybe this is just a semantics issue. If I roll a die, I'd say the events are just the numbers from $1$ to $6$. If you want to you can refer to, say, the event $4$ as "the set containing the number $4$". Is that what you meant or did you have something else in mind? – lulu Jan 12 '18 at 13:18
• I have literally described each event as a set! The event you are interest in, for example, is the third of mine...(Physics, $A$). The elements of that set are the subject, Physics, and the grade, $A$. I really can't guess what more you might want. – lulu Jan 12 '18 at 13:24
• Actually these pairs are the outcomes, so that indeed, one can choose as sample space the set $$\Omega=\{C,P\}\times\{A,\bar A\}=\{(C,A),(C,\bar A),(P,A),(P,\bar A)\}$$ then there are $16$ events, not just $4$... – Did Jan 12 '18 at 13:41

There are 2 criterion to get A in physics: You must take physics, and you must get an A. So the chance here will be $1/2 * 1/3 = 1/6$. The sample space here can be modeled as such: there is a 1 mark chemistry question, requiring full marks to get A. So you can get 0 or 1 mark. There is a 2 mark physics question, requiring full marks too to get an A. Here it can be clearly seen how the sample space is like.

• Reason for downvotes? – QuIcKmAtHs Jan 12 '18 at 14:05