A grocery manager discovered that on any given weekday 75% of the customer sales amount to more than $100. What is the probability that none of the first 3 sales will be more than $100? Treat the sales as independent events.

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29-10-2022
Mathematics

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A grocery manager discovered that on any given weekday 75% of the customer sales amount to more than $100. What is the probability that none of the first 3 sales will be more than $100? Treat the sales as independent events.

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JuliaunaV111330 JuliaunaV111330 · Answer 1 · 2022-10-29T16:45:36+02:00

Let's call C the event of a weekday having sales more than $100, so the probability of this event is given by P(C) = 0.75.

Let's call C~ the complementary event (the sales being less than $100). The probability of C~ is given by:

[tex]\begin{gathered} P(C)+P(C\text{\textasciitilde})=1 \\ 0.75+P(C\text{\textasciitilde})=1 \\ P(C\text{\textasciitilde})=1-0.75 \\ P(C\text{\textasciitilde})=0.25 \end{gathered}[/tex]

Then, if we have 3 sales, the probability of all 3 not having sales more than $100 is:

[tex]\begin{gathered} P=P(C\text{\textasciitilde})\cdot P(C\text{\textasciitilde})\cdot P(C\text{\textasciitilde}) \\ P=0.25\cdot0.25\cdot0.25 \\ P=0.015625 \end{gathered}[/tex]

So the probability is 0.0156 or 1.56%.