Question

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3. A survey is carried out on students and profesors at Kutatown University, asking them whether they prefer Wawa or Sheeta. (Wawa and Sheets are gustations with similar business models that originated in Pennsylvania) [12 points total] Gas Station Wawa Sheeta 441 315 29 15 Student Professor [1 point *) How many people were in the study? b) Based on the frequencies from the table, what is the probability that a randomly selected person from the study is a professor? 12 points) c) What is the probability that a randomly selected person from this study is either a student or prefers Sheetz, but not both? (3 points d) Find the probability that a randomly selected person is a professor, given that the person prefers 4 points se Who are likely to prefer Sheet students or professors? Justify your answer lwy computing appropriate probabilita

Answer 1

a) There are 441+29+315+15=800 people in the study.

b) Probability that a randomly selected person is a professor= $\small \frac{29+15}{800}=\frac{11}{200}$

c) Let A =the event that a randomly selected person is a student

B= the event that a randomly selected person prefers Sheetz.

Then P(A)=
441 + 315 800 189 -200
  
P(B) = ? 15 + 315 33 80080

Then the required probability is

P(A - B) + P(B - A) = P(AUB) - PAB) P(A) + P(B) - 2PA B ) =

Now

315 63 800 160

So The answer is

$\small P(A)+P(B)-2P(A\cap B)=\frac{189}{200}+\frac{33}{80}-\frac{2\times 63}{160} =\frac{57}{100}=0.57$

d) Let A =the event that a randomly selected person is a professor

B= the event that a randomly selected person prefers Wawa.

$\small P(B)=\frac{441+29}{800}=\frac{47}{80}$

$\small P(A\cap B)=\frac{29}{800}$

So the required probability is $\small P(A| B)=\frac{P(A\cap B)}{P(B)}=\frac{29/800}{47/80}=\frac{29}{470}$

e)

Let A1 =the event that a randomly selected person is a student.

Let A2 =the event that a randomly selected person is a professor.

Let B =the event that a randomly selected person prefers Sheetz.

The probability that a person prefers Sheetz given that he is a student is$\small P(B| A1)=\frac{P(B\cap A1)}{P(A1)}=\frac{315/800}{(441+315)/800} =\frac{5}{12}$

The probability that a person prefers Sheetz given that he is a professor is

$\small P(B| A2)=\frac{P(B\cap A2)}{P(A2)}=\frac{15/800}{(29+15)/800} =\frac{15}{44}<\frac{5}{12}=P(B|A1)$

This implies Students are more likely to prefer Sheetz.