The chances of selecting a student at random who failed math, science, and and english are 0.3125, 0.4375, and 0.375, respectively. The chance that the selected student failed all these subjects is 0.0625; while the chance that he did not fail any of these subjects is 0.225. If the selected student failed math, then the chance that he did not fail science is 0.56. If the selected student failed english, then the chance that he failed math is 0.3. $$\begin{gathered}\text{Given}: P(M)=0.3125\quad P(S)=0.4375 \quad P(E)=0.375\\ P(MSE)=0.0625 \quad P(M^cS^cE^c)=0.225\\ P(S^c|M)=0.56\quad P(M|E)=0.3\end{gathered}$$ I have included below the questions and my attempts to answer them.
1). What is the probability of selecting a student who failed english or science? $$ P(E \cup S)= P(S)+P(E)-P(ES)$$ I used De Morgan's on $P((MSE)^c)$ to find $P(M\cup S\cup E)$ then inclusion-exclusion to find $P(ES)$ and eventually got $P(E\cup S)=0.1625$
2). What is the probability of selecting a student who failed science and math but not english? $$P(S ME^c)$$ Here, I used definition of set difference then property of probability function to get $P(SME^c)=P(SM)-P(SME)=0.075$ $$$$ 3). If the selected student did not fail english, what is the probability that he failed math or did not fail science? $$ P(M\cup S^c|E^c)$$ For this one, I don't know how to get $P((M\cup S^c)\cap E^c)$. Please let me know how I can start on this and if I made any mistakes above.
I like to solve these exercises using Venn's diagrams.
After a little brainstorming you get this diagram
which shows you the answer to any question they can put you. Numbers are expressed in %
In the particular case, your answers are the following
$$77.5-12.5=65\%$$
$$7.5\%$$
$$22.5+12.5+7.5=42.5\%$$