Given that $$P(A) = 0.35 \quad P(B) = 0.45 \quad \text{and} \quad P(A \cap B) = 0.13$$ find (a) $ P(A' | B') $ (2) (b) Explain why the events A and B are not independent. (1) The event C has $ P(C) = 0.20 $ The events A and C are mutually exclusive and the events B and C are statistically independent. (c) Draw a Venn diagram to illustrate the events A, B and C, giving the probabilities for each region. (5) (d) Find $ P(B \cup C') $ (2)

A-Level Edexcel Maths Statistics Basic Probability: Given that $$P(A) = 0.35 \quad

Given that $$P(A) = 0.35 \quad P(B) = 0.45 \quad \text{and} \quad P(A \cap B) = 0.13$$ find (a) $ P(A' | B') $ (2) (b) Explain why the events A and B are not independent - Edexcel - A-Level Maths Statistics - Question 4 - 2017 - Paper 2

Question 4

$Given-that--$$P(A)-=-0.35-\quad-P(B)-=-0.45-\quad-\text{and}-\quad-P(A-\cap-B)-=-0.13$$--find--(a)-$-P(A'-|-B')-$-(2)--(b)-Explain-why-the-events-A-and-B-are-not-independent-Edexcel-A-Level Maths Statistics-Question 4-2017-Paper 2.png$

Given that $$P(A) = 0.35 \quad P(B) = 0.45 \quad \text{and} \quad P(A \cap B) = 0.13$$ find (a) $ P(A' | B') $ (2) (b) Explain why the events A and B are not i... show full transcript

Worked Solution & Example Answer:Given that $$P(A) = 0.35 \quad P(B) = 0.45 \quad \text{and} \quad P(A \cap B) = 0.13$$ find (a) $ P(A' | B') $ (2) (b) Explain why the events A and B are not independent - Edexcel - A-Level Maths Statistics - Question 4 - 2017 - Paper 2

Step 1

Find $ P(A' | B') $

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Answer

To find ( P(A' | B') ), we can use the formula:

$P(A' | B') = 1 - P(A | B)$

First, we find ( P(A | B) ) using:

$P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{0.13}{0.45} \approx 0.2889$

Now, substituting into our original formula:

$P(A' | B') = 1 - 0.2889 \approx 0.7111$

Step 2

Explain why the events A and B are not independent.

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Answer

Events A and B are not independent because:

$P(A \cap B) \neq P(A) \cdot P(B)$

Calculating, we find:

$P(A \cap B) = 0.13 \quad \text{and} \quad P(A) \cdot P(B) = 0.35 \cdot 0.45 = 0.1575$

Since ( 0.13 \neq 0.1575 ), the events are dependent.

Step 3

Draw a Venn diagram to illustrate the events A, B and C.

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Answer

In the Venn diagram:

Region for A only contains 0.22
Region for B only contains 0.09
Region for C only has 0.20
The overlap between A and B contains 0.13
The area A and C overlaps shows 0.00 since they are mutually exclusive.

Each region should be clearly labeled to show these probabilities.

Step 4

Find $ P(B \cup C') $

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Answer

Using the formula for the union of two probabilities:

$P(B \cup C') = P(B) + P(C') - P(B \cap C')$

We know:

( P(B) = 0.45 )
( P(C') = 1 - P(C) = 1 - 0.20 = 0.80 )
To find ( P(B \cap C') ), we use the independence state of B and C:

Thus:

$P(B \cap C') = P(B) \cdot P(C') = 0.45 \cdot 0.80 = 0.36$

Calculating:

$P(B \cup C') = 0.45 + 0.80 - 0.36 = 0.89$

Given that $$P(A) = 0.35 \quad P(B) = 0.45 \quad \text{and} \quad P(A \cap B) = 0.13$$ find (a) \( P(A' | B') \) (2) (b) Explain why the events A and B are not independent - Edexcel - A-Level Maths Statistics - Question 4 - 2017 - Paper 2

Worked Solution & Example Answer:Given that $$P(A) = 0.35 \quad P(B) = 0.45 \quad \text{and} \quad P(A \cap B) = 0.13$$ find (a) \( P(A' | B') \) (2) (b) Explain why the events A and B are not independent - Edexcel - A-Level Maths Statistics - Question 4 - 2017 - Paper 2

Find \( P(A' | B') \)

Explain why the events A and B are not independent.

Draw a Venn diagram to illustrate the events A, B and C.

Find \( P(B \cup C') \)

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