A sample of 200 households was obtained from a small town. Each household was asked to complete a questionnaire about their purchases of takeaway food. A is the event that a household regularly purchases Indian takeaway food. B is the event that a household regularly purchases Chinese takeaway food. It was observed that P(B | A) = 0.25 and P(A | B) = 0.1. Of these households, 122 indicated that they did not regularly purchase Indian or Chinese takeaway food. A household is selected at random from those in the sample. Find the probability that the household regularly purchases both Indian and Chinese takeaway food.

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A sample of 200 households was obtained from a small town - AQA - A-Level Maths Mechanics - Question 15 - 2019 - Paper 3

Question 15

A sample of 200 households was obtained from a small town. Each household was asked to complete a questionnaire about their purchases of takeaway food. A is the even... show full transcript

Worked Solution & Example Answer:A sample of 200 households was obtained from a small town - AQA - A-Level Maths Mechanics - Question 15 - 2019 - Paper 3

Step 1

Find P(A) and P(B)

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Answer

We know that:

From the conditional probability, we have: $P(B | A) = \frac{P(A \cap B)}{P(A)}$ which gives us: $P(A \cap B) = P(B | A) \times P(A)$
From the other conditional probability: $P(A | B) = \frac{P(A \cap B)}{P(B)}$ which gives: $P(A \cap B) = P(A | B) \times P(B)$
Let P(A) be represented as $x$ and P(B) as $y$ . Therefore:
- From the first condition:
  $P(A \cap B) = 0.25x$
- From the second condition:
  $P(A \cap B) = 0.1y$
Let us name both equal values: $0.25x = 0.1y$ . Rearranging gives us: $y = 2.5x$ .

Step 2

Find total households that purchase neither

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Answer

From the total of 200 households, we know that 122 do not purchase Indian or Chinese takeaway food. Therefore, the number of households purchasing either Indian or Chinese takeaway food is:

$200 - 122 = 78$ .

Step 3

Use Addition Law of Probability

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Answer

Using the addition law of probability:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Where:

$P(A \cup B) = 78/200$
Substitute $P(B)$ using $y = 2.5x$ :
$P(B) = \frac{2.5x}{200}$
Substitute $P(A)$ :
$P(A) = \frac{x}{200}.$

Putting these values in:

$\frac{78}{200} = \frac{x}{200} + \frac{2.5x}{200} - \frac{0.1y}{200}$ . On solving the above, we can find $x$ and subsequently $y$ .

Step 4

Calculate P(A \cap B)

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Answer

Substituting back to find:

Substitute values into earlier established expressions:
- For $P(A \cap B)$ , we can compute: $P(A \cap B) = 0.25x = 0.1 \cdot 2.5x = 0.25x.$
As both establish the same, we can conclude: $P(A \cap B) = \frac{39}{200}.$

Thus, the probability that the household regularly purchases both Indian and Chinese takeaway food is: $P(A \cap B) = \frac{39}{200}$ . This fraction can also be expressed in decimal as: 0.195.

A sample of 200 households was obtained from a small town - AQA - A-Level Maths Mechanics - Question 15 - 2019 - Paper 3

Worked Solution & Example Answer:A sample of 200 households was obtained from a small town - AQA - A-Level Maths Mechanics - Question 15 - 2019 - Paper 3

Find P(A) and P(B)

Find total households that purchase neither

Use Addition Law of Probability

Calculate P(A \cap B)

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