Respuesta :
Answer and explanation:
Given : At Tech Express, 40% of the customers purchases small coffee and 60% purchases medium. Of those customers purchasing small coffee, 30% prefer decaf. Of those purchasing medium coffee, 50% prefer decaf.
Let [tex]A_1[/tex] customer purchase small coffee.
i.e. [tex]P(A_1)=40\%=0.4[/tex]
[tex]A_2[/tex] customer purchase medium coffee.
i.e. [tex]P(A_2)=60\%=0.6[/tex]
Let B be the customer purchase prefer decaf.
So, [tex]P(B|A_1)=30\%=0.3[/tex]
[tex]P(B|A_2)=50\%=0.5[/tex]
(a) What is the probability that the next customer will request medium and decaf coffee?
i.e. [tex]P(A_2\cap B)=P(A_2)\times P(B|A_2)[/tex]
[tex]P(A_2\cap B)=0.6\times 0.5[/tex]
[tex]P(A_2\cap B)=0.3[/tex]
(b) What is the probability that the next customer prefers decaf?
i.e. [tex]P(B)=P(A_1\cap B)+P(A_2\cap B)[/tex]
[tex]P(B)=P(A_1)\times P(B|A_1)+P(A_2)\times P(B|A_2)[/tex]
[tex]P(B)=0.4\times 0.3+0.6\times 0.5[/tex]
[tex]P(B)=0.12+0.3[/tex]
[tex]P(B)=0.42[/tex]
(c) If the next customer prefers decaf, what is the probability that small is requested?
i.e. [tex]P(A_1|B)=\frac{P(A_1\cap B)}{P(B)}[/tex]
[tex]P(A_1|B)=\frac{0.12}{0.42}[/tex]
[tex]P(A_1|B)=0.28[/tex]
