Terence owns a local shop. His shop has three checkouts, at least one of which is always staffed. A regular customer observed that the probability distribution for $N$, the number of checkouts that are staffed at any given time during the spring, is $$P(N = n) = \frac{3}{4} \left( \frac{1}{4} \right)^{n-1} \text{ for } n = 1, 2\n k \text{ for } n = 3$$ 11 (a) Find the value of k. 11 (b) Find the probability that a customer, visiting Terence's shop during the spring, will find at least 2 checkouts staffed.

A-Level AQA Maths Pure Geometric Sequences & Series: Terence owns a local shop. His s

Terence owns a local shop - AQA - A-Level Maths Pure - Question 11 - 2019 - Paper 3

Question 11

Terence owns a local shop. His shop has three checkouts, at least one of which is always staffed. A regular customer observed that the probability distribution for ... show full transcript

Worked Solution & Example Answer:Terence owns a local shop - AQA - A-Level Maths Pure - Question 11 - 2019 - Paper 3

Step 1

Find the value of k.

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Answer

To find the value of $k$ , we need to use the property that the total probability must sum to 1:

$P(N = 1) + P(N = 2) + P(N = 3) = 1$

Let's calculate:

For $n = 1$ :
$P(N = 1) = \frac{3}{4} \left( \frac{1}{4} \right)^{1-1} = \frac{3}{4}$
For $n = 2$ :
$P(N = 2) = \frac{3}{4} \left( \frac{1}{4} \right)^{2-1} = \frac{3}{16}$

Now substituting these into the total:

$\frac{3}{4} + \frac{3}{16} + k = 1$

To combine the fractions: $\frac{12}{16} + \frac{3}{16} + k = 1$
$\frac{15}{16} + k = 1$
Thus,
$k = 1 - \frac{15}{16} = \frac{1}{16}$

Step 2

Find the probability that a customer, visiting Terence’s shop during the spring, will find at least 2 checkouts staffed.

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Answer

To find the probability that at least 2 checkouts are staffed, we calculate: $P(N \geq 2) = P(N = 2) + P(N = 3)$

From part (a), we established:

$P(N = 2) = \frac{3}{16}$
For $n = 3$ :
$P(N = 3) = k = \frac{1}{16}$

Now, substitute these values: $P(N \geq 2) = \frac{3}{16} + \frac{1}{16} = \frac{4}{16} = \frac{1}{4}$

Thus, the final answer is: $P(N \geq 2) = \frac{1}{4}$

Terence owns a local shop - AQA - A-Level Maths Pure - Question 11 - 2019 - Paper 3

Worked Solution & Example Answer:Terence owns a local shop - AQA - A-Level Maths Pure - Question 11 - 2019 - Paper 3

Find the value of k.

Find the probability that a customer, visiting Terence’s shop during the spring, will find at least 2 checkouts staffed.

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