Question-6

Out of the students in a class, \(60\%\) do well in mathematics, \(70\%\) do well in physics and \(50\%\) do well in both. Find the probability that a randomly selected student doesn’t do well in both mathematics and physics.

Hint

Define \(A\) and \(B\) as the following events:

• \(A\): student does well in mathematics
• \(B\): student does well in physics

What event are we interested in?

Answer

\(0.2\)

Solution

We have the following events:

• \(A\): student does well in maths
• \(B\): student does well in physics
• \(P(A^{c})\): student doesn’t do well in maths
• \(P(B^{c})\): student doesn’t do well in physics
• \(P(A^{c} \cap B^{c})\): student doesn’t do well both maths and physics

Using DeMorgan’s laws, we have: \[ A^{c} \cap B^{c} = (A \cup B)^{c} \] This gives us: \[ \begin{aligned} P(A^{c} \cap B^{c}) &= P(A \cup B)^{c}\\ &= 1 - P(A \cup B)\\ \end{aligned} \] We can now compute \(P(A \cup B)\): \[ \begin{aligned} P(A \cup B) &= P(A) + P(B) - P(A \cap B)\\ &= 0.6 + 0.7 - 0.5\\ &= 0.8 \end{aligned} \] The required probability is therefore \(\boxed{0.2}\).