In an examination, at least 70% of the students failed in physics, at least 72% failed in chemistry, at least 80% failed in mathematics and at least 85% failed in english. How many at least must have failed in all the four subjects.(all information is sufficient to calculate the answer)
Well, I don't understand what does that 'at least failed' means. And how would you know the exact number of students failing/passing in each subject and then calculating the students failing in all four from venn diagram which would be theoretically very difficult. Please help me!
Given: $$\begin{cases}P(Ph)\ge 0.7,\\ P(Ch)\ge 0.72,\\P(M)\ge 0.8,\\ P(E)\ge 0.85 \end{cases}$$ we get: $$P(Ph\cap Ch)=P(Ph)+P(Ch)-P(Ph\cup Ch)\ge P(Ph)+P(Ch)-1\ge 0.42;\\ P(M\cap E)=P(M)+P(E)-P(M\cup E)\ge P(M)+P(E)-1\ge 0.65;\\ P((Ph\cap Ch)\cap(M\cap E))=P(Ph\cap Ch)+P(M\cap E)-P(Ph\cup Ch\cup M\cup E)=\\ P(Ph\cap Ch)+P(M\cap E)-1\ge 0.42+0.65-1=0.07. $$ Similar post.