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Two events, A and B, are represented in the diagram - Leaving Cert Mathematics - Question b - 2016

Question b

Two events, A and B, are represented in the diagram. P(A ∩ B) = 0.1, P(B | A) = 0.3 and P(A | B) = x. Write P(A) in terms of x and hence, or otherwise, find the va... show full transcript

Worked Solution & Example Answer:Two events, A and B, are represented in the diagram - Leaving Cert Mathematics - Question b - 2016

Step 1

Write P(A) in terms of x

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Answer

Using the formula for conditional probability:

$P(A | B) = \frac{P(A \cap B)}{P(B)}$

We rearrange it as follows:

$P(A \cap B) = P(A | B) \times P(B)$

Substituting the given values:

$0.1 = x \times 0.4$

From this, we can find P(A):

$P(A) = x + 0.1$

Step 2

Find the value of x for which the events A and B are independent

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Answer

For events A and B to be independent, the following condition must hold:

$P(A \cap B) = P(A) \times P(B)$

Substituting the values into this condition:

$0.1 = (x + 0.1) \times 0.4$

Expanding this gives:

$0.1 = 0.4x + 0.04$

Rearranging the equation:

0.4x = 0.06 \\ x = \frac{0.06}{0.4} = 0.15$$

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