Question
A and B are two events such that P(A) = 0.5, P(B) = 0.7 and P(A ∪ B) = 0.85.
(a) Find P(A ∩ B).
(b) Determine whether events A and B are independent.
Answer/Explanation
Ans:
(a) Using the combined events formula, we get
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
0.85 = 0.5 + 0.7 – P(A ∩ B)
P(A ∩ B) = 0.5 +0.7 – 0.85
P(A ∩ B) = 0.35
(b) Using the conditional probability formula, we have
\(P(A|B)=\frac{P(A\cap B)}{P(B)}\)
\(=\frac{0.35}{0.7}\)
= 0.5
= P(A)
Hence, A and B are independent
Question
The following diagram shows part of the graph of a quadratic function f.
The vertex is at point A and the x-intercepts are at points B and C.
The function f can be written in the form f(x) = (x – h) 2 + k.
(a) Write down the values of h and k.
The function f can also be written in the form f(x) = (x + p) (x – q).
(b) Write down the values of p and q.
(c) Find the y-intercept of the graph of f.
Answer/Explanation
Ans:
(a) h = 2 and k = -9
(b) p = 1 and q = 5
(c) Evaluating f(x) = (x + 1) (x – 5) for x = 0, we have
f(0) = (0 +1) (0-5)
= (1)(-5)
=-5
Hence the point D(0, -5) is the y-intercept of the graph of f.