IBDP Maths SL 4.11 Formal definition and use of the formulae AA HL Paper 1- Exam Style Questions- New Syllabus

(a)
The probability of the union can be expressed as \( P(A \cup B) = P(A \cap B’) + P(B) \).
Substituting the known values: \( \dfrac{5}{8} = \dfrac{7}{24} + P(B) \).
Rearranging to find \( P(B) \):
\( P(B) = \dfrac{15}{24} – \dfrac{7}{24} = \dfrac{8}{24} = \dfrac{1}{3} \).
\( \boxed{P(B) = \frac{1}{3}} \)

(b)
If \( A \) and \( B \) are independent, then \( P(A \cap B’) = P(A) \cdot P(B’) \).
Calculate \( P(B’) \): \( P(B’) = 1 – P(B) = 1 – \dfrac{1}{3} = \dfrac{2}{3} \).
Now determine \( P(A) \):
\( P(A) \cdot \dfrac{2}{3} = \dfrac{7}{24} \implies P(A) = \dfrac{7}{24} \cdot \dfrac{3}{2} = \dfrac{7}{16} \).
Then \( P(A’) = 1 – P(A) = 1 – \dfrac{7}{16} = \dfrac{9}{16} \).
For independent events, \( P(A’ \mid B) = P(A’) \):
\( P(A’ \mid B) = \dfrac{9}{16} \).
\( \boxed{P(A’ \mid B) = \frac{9}{16}} \)