Question

Of the 75 employees in the office building in dc, 40 had taken the subway to work sometime during the past year, 22 had taken the bus sometime during the past year, and 55 had taken either the subway or the bus or both to work last year. If one person is selected at random, show you work and find the following:

a) The probability that the person had taken both the subway and the bus.

b) The probability that the person had taken the bus but not the subway.

c) The probability that the person had taken neither the bus nor the subway

d) The probability that the person had not taken both the bus and the subway

Answer 1

a) P(Both) = 7/75

b) P(B&~S) = 1/5

c) P(~B&~S) = 4/15

d) P(~Both) = 68/75

Explanation:

First, the 40 subway riders includes SubwayOnly people and also Subway&Bus...BOTH people.

Likewise, 22 bus people are BusOnly people as well as Bus&Subway--BOTH people.

So there is some overlap and we should sort that out before writing these probabilities. Let's use a Venn diagram.

40+22 is 62, but we know theres only 55 public transportation people. 62-55 is 7. Thats the overlap. There are 7 people who ride BOTH bus and subway. We put them in the middle and subtract to find the BusOnly and SubwayOnly people.

40 -7 is 33 SubwayOnly people.

22 - 7 is 15 BusOnly people.

55 people ride some public transpo, so 75 - 55 is 20 who people ride nothing (so cars or walkers).

Now we can write probabilities.

a) BOTH is 7/75

b) BusOnly is 15 people so probability is 15/75, which can be reduced so 15/75 = 1/5

c) NoBus and NoSub people are the 20 car or walkers. 20/75 = 4/15

NotBOTH people are the 20 (car/walkers) also the BusOnly and SubwayOnly. Thats 20+33+15, 68 people. Another way to think of it is ALL - 7 =

75 - 7 =

68 people, so the probability is 68/75.

also see image.

Answer 2

a)11/15

b)22/75

c)0/75

d)66/75

Explanation:

a) Altogether, 55 people took the bus or the subway out of the 75 employees. As a fraction, this is 55/75 which can be simplified into 11/15

b) 22 out of the 75 people took the bus but not the subway. As a fraction, this is 22/75

c) Everyone took either the subway or the bus or both in the past year, meaning 0 out of 75 people took neither the subway, bus, or both. As a fraction, this is 0/75

d) The people who took the bus and the subway but not both would be 40+22 out of 75. As a fraction, this is 66/75

I hope this helps!