A telephone solicitor is responsible for canvassing three suburbs. In the past, 60% of the completed calls to Belle Meade have resulted in contributions, compared to 55% for Oak Hill and 35% for Antioch. Her list of telephone numbers includes one thousand households from Belle Meade, one thousand from Oak Hill, and two thousand from Antioch. Suppose that she picks a number at random from the list and places the call. What is the probability that she gets a donation? Larsen, Richard J.. An Introduction to Mathematical Statistics and Its Applications (p. 44). Pearson Education. Kindle Edition.

The probablem states that her list of telephone numbers includes one thousand households from Belle Meade, one thousand from Oak Hill, and two thousand from Antioch. In all, she has 4000 total telephone numbers.

She has 1000 numbers from Belle Maede. So there is a 1000/4000 = 25% probability that she calls a person from Belle Maede.

She has 1000 numbers from Oak Hill. So there is a 1000/4000 = 25% probability that she calls a person from Oak Hill.

She has 2000 numbers from Antioch. So there is a 2000/4000 = 50% probability that she calls a person from Antioch.

What is the probability that she gets a donation?

[tex]P = P_{1} + P_{2} + P_{3}[/tex]

[tex]P_{1}[/tex] is the probability the she randomsly choses a person from Belle Maede and receives a donation. So:

[tex]P_{1} = 0.6*0.25 = 0.15[/tex]

[tex]P_{2}[/tex] is the probability the she randomsly choses a person from Oak Hill and receives a donation. So:

[tex]P_{2} = 0.55*0.25 = 0.1375[/tex]

[tex]P_{3}[/tex] is the probability the she randomsly choses a person from Antioch and receives a donation. So:

[tex]P_{2} = 0.35*0.50 = 0.175[/tex]

[tex]P = P_{1} + P_{2} + P_{3} = 0.15 + 0.1375 + 0.175 = 0.4625[/tex]

There is a 46.25% probability that she receives a donation.