Question

Four tourists plan on taking a bus tour around Toronto. This tour allows them to hop on and off the bus at any of the 8 stops available. The available stops are the CN tower, Harbour Front, Queen’s Quay, Dundas Square, Casa Loma, Distillery District, Bata Shoe Museum and the Royal Ontario Museum.

1. What is the probability that two attend one location and two attend another same location?
2. What is the probability that no tourists hop off at the Distillery District and Bata Shoe Museum?
3. What is the probability that at least one tourist gets off at Casa Loma?

Answer 1

Solution

Back-up Theory

Probability of an event E, denoted by P(E) = n/N ……………..….................……………(1)

where

n = n(E) = Number of outcomes/cases/possibilities favorable to the event E and

N = n(S) = Total number all possible outcomes/cases/possibilities.

Now to work out the solution,

There are 4 tourists and 8 locations. Since the same location can be visited by

more than one tourist, total number of possible location visits = 8⁴ = 4096 ................. (2)

Hence, vide (1), N = 4096 ..............................................................................................(3)

Part (a)

Two attending one location can any 2 out of 4 tourists in ⁴C₂ = 6 ways. Then, the two necessarily visit another location. Thus, there is only one possibility.

Now, the first pair can choose any one of 8 locations in 8 ways and then the second

pair can choose only one of the remaining 7 locations.

So, total number of possibilities = 6 x 8 x 7 = 336 and hence vide (1), n = 336 ............ (4)

Finally, vide (1), (3) and (4),

Probability that two attend one location and two attend another location = 336/4096

= 0.0820 Answer 1

Part (b)

If no tourists hop off at the Distillery District and Bata Shoe Museum, the number of

location choices reduces from 8 to 6. Hence, total number of possible location visits

= 6⁴ = 1296 and this then becomes n vide (1)................................................................. (5)

Thus, vide (1), (3) and (5),

Probability that no tourist hop off at the Distillery District and Bata Shoe Museum

= 1296/4096

= 0.3164   Answer 2

Part (c)

Probability that at least one tourist gets off at Casa Loma

= 1 –P(no tourist gets off at Casa Loma)

No tourist gets off at Casa Loma => the number of location choices reduces from

8 to 7. Hence, total number of possible location visits = 7⁴ = 2401 and this then

becomes n vide (1)........................................................................................................... (6)

Thus, vide (1), (3) and (6),

Probability that at least one tourist gets off at Casa Loma

= 1 – (2401/4096)

= 1 – 0.5862

= 0.4138 Answer 3

DONE