Question

In a small country there are 3 supermarkets: P, Q and R. 60% of the population shop at P, 36% shop at Q, 34% shop at R, 18% shop at P and Q, 15% shop at P and R, 4% shop at q and R, and 2% shop at all 3 supermarkets. A person is selected at random. Determine the probability that the person shops at:
d) either P or Q
e) P, given that the person shops at least one supermarket
f) R, given that the person shops at either P or Q or both.

Answer 1

Given :- In a small country there are 3 supermarkets: P, Q and R. 60% of the population shop at P, 36% shop at Q, 34% shop at R, 18% shop at P and Q, 15% shop at P and R, 4% shop at q and R, and 2% shop at all 3 supermarkets. A person is selected at random.

To Find :-

• P, given that the person shops at least one supermarket .
• R, given that the person shops at either P or Q or both.

Answer :-

given that,

• 2% shop at all 3 supermarkets .
• Total = 100% .

so,

→ shop at P and R, but not Q = 15% - 2% = 13%

→ shop at P and Q, but not R = 18% - 2% = 16%

→ shop at Q and R, but not P = 4% - 2% = 2%

→ shop at P only = 60% - (18% + 15%) + 2% = 29%

→ shop at Q only = 36% - (18% + 4%) + 2% = 16%

→ shop at R only = 34% - (15% + 4%) + 2% = 17%

→ shop at none of the supermarkets = Total - [(P only + Q only + R only) + {(P and R) + (P and Q) + (Q and R)} + shop at all three] = 100% - [(29% + 16% + 17%) + (13% + 16% + 2%) + 2%] = 100% - (62% + 31% + 2%) = 100% - 95% = 5% .

then,

→ Person shop at atleast one supermarket = Total - shop at none of the supermarkets = 100% - 5% = 95% .

therefore,

→ P = 95% / 100% = 0.95 (Ans.)

now,

→ The person shops at either P or Q = 29% + 16% - 16% = 29% = 29% .

→ The person shops at P and Q both = 16% .

hence,

→ R = (29% + 16%) / 100% = (45/100)% = 0.45 (Ans.)

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Answer 2

Answer:

d) $\frac{39}{50\\}$     e)$\frac{12}{19}$    f) $\frac{17}{78}$

Step-by-step explanation:

d) $\frac{29+16+2+13+2+16}{100} =\frac{78}{100} =\frac{39}{50}$

e) P(P I at at least one super market) = $\frac{\frac{29+16+2+13}{100} }{\frac{19}{20} } = \frac{\frac{60}{100} }{\frac{19}{20} } =\frac{\frac{3}{5} }{\frac{19}{20} } = \frac{12}{19}$

f) P (R I P∩Q) = $\frac{13+2+2}{13+2+2+16+16+29} = \frac{17}{78}$

Note: The picture lacks 5% of the population who do not shop in any of the supermarkets. (those who are not in any set)

I forgot to add :)