I will present two problems alongside solutions, student is doing problems of type I like a cakewalk but has several issues with the problems of type 2;
Type I
Consider an experiment of rolling two dice:
Sample space $$ S = \{(1,1),(1,2),(1,3), \cdots, (6,6) \} $$
Let $A$ be the event of getting 6 as sum on two dice:
Event $$A = \{(1,5),(2,4),(3,3),(4,2),(5,1)\}$$
Let $B$ be the event of getting 4 on first die:
Event $$B = \{(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)\}$$
Now, the probability of getting sum of 6 on two dice given that first die appears as 4 is given by
$$p(A \mid B) = \dfrac{p(A\cap B)}{p(B)} = \dfrac{p\{(4,2)\}}{p(B)} = \dfrac{1}{6}$$
Type II
Let $C$ be the event of having cancer and $p(C) = 1/2$
The probability of having both tumor and cancer is $p(C\cap T) = 1/6$
Then the probability of having tumor given cancer is given by
$$p(T\mid C) = \dfrac{p(T\cap C)}{p(C)} = \dfrac{1}{3}$$
Students are understanding type I problems and type 2 problems well, but some students are asking to present $C, T, C\cap T$ in terms of sets. I tried to convince them using Venn diagrams. But they are asking for either roster form (for discrete sets) or set builder form(for any set).
I am trying to do like follows
\begin{align} \text{Sample space} = S &= \{x \mid \text{$x$ is a living being} \} \\ C &= \{x \mid \text{$x$ has cancer}\} \\ C\cap T &= \{x \mid \text{$x$ has both tumor and cancer}\} \\ \end{align}
Is it right way to do? Students are facing difficulty and they are asking every event inform of set, since definition of event is that it is a subset of sample space (which is a set).
primary-education
is certainly wrong here, no? $\endgroup$