Question

Question 4 [3 marks]

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. [1 mark] What is the probability that two attend one location and two attend another same location?
2. [1 mark] What is the probability that no tourists hop off at the Distillery District and Bata Shoe Museum?
3. [1 mark] 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 favourable 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