In a class of 30 students, 20 study Physics, 6 study Biology and 4 study both Physics and Biology - Leaving Cert Mathematics - Question b - 2013

To investigate if events E (the student studies Physics) and F (the student studies Biology) are independent, we apply the definition of independent events:

1. Calculating P(E and F): This represents the probability that a student studies both Physics and Biology. From the Venn Diagram, there are 4 students studying both subjects out of 30 total students.

   $P(E \cap F) = \frac{4}{30} = \frac{2}{15}$

2. Calculating P(E): This is the probability that a student studies Physics.

   • From our calculations:

   $P(E) = \frac{20}{30} = \frac{2}{3}$

3. Calculating P(F): This is the probability that a student studies Biology.

   • From our earlier calculations:

   $P(F) = \frac{6}{30} = \frac{1}{5}$

4. Calculating P(E) x P(F):

   $P(E) \times P(F) = \left(\frac{2}{3}\right) \times \left(\frac{1}{5}\right) = \frac{2}{15}$

5. Comparison:

   • Now, we check if $P(E \cap F) = P(E) \times P(F)$:

   Since: $P(E \cap F) = \frac{2}{15}$ and $P(E) \times P(F) = \frac{2}{15}$ We conclude:

   Events E and F are independent events.