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

To find the value of r, we note that:

$r = P(A ∩ B) - p$

Given that:

• P(A ∩ B) = P(A) + P(B) - P(A ∪ B)
• We need to calculate P(A ∩ B) using:
• Total Probability:

$P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.5 + 0.6 - 0.28 = 0.78$

Then,

$r = 0.5 + 0.6 - 0.28 = 0.82 - 0.15 = 0.22$. Therefore, the value of r is 0.22.

To find the values of s and t, we can use:

1. The total probability of all events in the Venn diagram should sum up to 1:

   $s = P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - (p + q + r)$. Since we have already calculated the values:

   • p = 0.15, q = 0.10, r = 0.22:

   $s = 0.5 + 0.6 + 0.25 - (0.15 + 0.10 + 0.22) = 0.5 + 0.6 + 0.25 - 0.47 = 0.28$.

2. Next, we can find t by using:

   $t = P(B) - s = 0.6 - 0.28 = 0.28$.

Events A and B are independent if the following holds:

$P(A ∩ B) = P(A) imes P(B)$

Calculating:

• We have:

$P(A) imes P(B) = 0.5 imes 0.6 = 0.30$

However, from our calculations,

$P(A ∩ B) = r + p = 0.22 + 0.15 = 0.37$.

This value is not equal to 0.30, hence:

Conclusion: Events A and B are not independent.

To find the conditional probability P(B | A ∪ C), we can use the formula:

$P(B | A ∪ C) = \frac{P(B ∩ (A ∪ C))}{P(A ∪ C)}$

1. Find P(A ∪ C):

   $P(A ∪ C) = P(A) + P(C) - P(A ∩ C)$

Given:

• P(A ∩ C) = s = 0.28, thus we calculate:

  $P(A ∪ C) = 0.5 + 0.25 - 0.28 = 0.47$.

2. Now calculate P(B ∩ (A ∪ C)):

Using values for A and intersection with B, can come from entries:

$= 0.22 + 0.15 = 0.37$

Substituting:

$P(B | A ∪ C) = \frac{0.37}{0.47} \\approx 0.79$.