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 $$P(N = n) = \begin{cases} \frac{3}{4} \left( \frac{1}{4} \right)^{n-1} & \text{for } n = 1, 2 \\ k & \text{for } n = 3 \end{cases}$$

A-Level AQA Maths Mechanics Quantities, Units & Modelling: Terence owns a local shop. His s

Terence owns a local shop - AQA - A-Level Maths Mechanics - 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 Mechanics - 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 ensure that the total probability sums up to 1. We have:

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

Substituting the given probabilities:

$\frac{3}{4} \left( \frac{1}{4} \right)^0 + \frac{3}{4} \left( \frac{1}{4} \right)^1 + k = 1$

Calculating the values:

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

Now turning all terms to fractions having a common denominator of 16:

$\frac{12}{16} + \frac{3}{16} + k = 1$

Thus:

$\frac{15}{16} + k = 1$

Rearranging gives:

$k = 1 - \frac{15}{16} = \frac{1}{16}$

Step 2

Select relevant probability for $N \geq 2$ checkouts staffed

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Answer

We need to find the probability that at least 2 checkouts are staffed. This is given by:

$P(N \geq 2) = P(N = 2) + P(N = 3)$

Using our earlier results:

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

And since we found that $k = \frac{1}{16}$ for $P(N = 3)$ , we have:

$P(N = 3) = k = \frac{1}{16}$

Thus combining both:

$P(N \geq 2) = \frac{3}{16} + \frac{1}{16} = \frac{4}{16} = \frac{1}{4}$

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

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

Find the value of $k$

Select relevant probability for $N \geq 2$ checkouts staffed

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