The Venn diagram shows three events A, B and C, where p, q, r, s and t are probabilities - Edexcel - A-Level Maths Statistics - Question 3 - 2017 - Paper 1

Using the information from the Venn diagram, we note:

$r = P(A) - (p + s)$

From the given data, we substitute:

$r = 0.5 - (0.15 + 0.08) = 0.5 - 0.23 = 0.22$

Using the relationships among the probabilities:

We already found:

• $s = P(A ext{ and } B) = 0.08$
• $t = P(B) - (s + q) = 0.6 - (0.08 + 0.10) = 0.6 - 0.18 = 0.42$

To check if A and B are independent, we verify if:

$P(A ext{ and } B) = P(A) imes P(B)$

Given that: $P(A ext{ and } B) = s = 0.08 \quad ext{and} \quad P(A) imes P(B) = 0.5 imes 0.6 = 0.3$

Since $0.08 \neq 0.3$, A and B are not independent.

Using the definition of conditional probability:

P(B | A igcup C) = \frac{P(B ext{ and } (A igcup C))}{P(A igcup C)}

We can calculate:

1. P(A igcup C) = P(A) + P(C) - P(A ext{ and } C) = 0.5 + 0.25 - (r + p) = 0.5 + 0.25 - (0.22 + 0.15) = 0.58.

2. To find P(B ext{ and } (A igcup C)), we know: $P(B) = 0.6$

Thus, we need to calculate: P(B | A igcup C) = \frac{0.6 - 0.08}{0.58} = \frac{0.52}{0.58} \\ \approx 0.896.