Problem 4

PROBLEM 4: PARKING LOT PROBLEM (3 points possible) Mary and Tom park their cars in an empty parking lot with parking spaces (i.e,

n≥2 consecutive

w here only one car fits in each n spaces in a row, where

space). Mary and Tom pick parking spaces at random. (All pairs of parking spaces are equally likely.) What is the probability that there is at a t most one empty parking space between them? (Express your answer using standard notation..) notation

(4*n-6)/(n*(n-1))

(n−1)⋅(n−2) Answer:

The sample space is

(i, j  j)

outcome

Ω={(i, j  j):i≠ j,1≤i, j  j≤n}

, where

 indicates that Mary and Tom parked in slots

i  and  j ,

respectivel respec tively. y. We apply apply the discrete discrete uniform uniform probabilit probability y law to find the the required requi red probabilit probability. y. We are intereste interested d in the probabili probability ty of the event event

 A={(i, j  j)∈Ω:|i− j|≤2}. We first find the cardinality of the set of

Ω

Ω  is

Ω

. There are

 excludes outcomes of the form

n2−n=n(n−1) .

butt sin inc ce n2  pairs (i, j  j) , bu

(i,i)

, the the ca card rdin inal alit ity y

If

n≥3 , event  A  consists of the four lines indicated in the figure

above and contains event

2(n−1)+2(n−2)=4n−6

 elements. If

n=2 ,

 A  contains exactly 2 elements, namely, (1,2)  and (2,1) ,

which agrees with the formula

4(2)−6=2

. Therefore,

P( A)=4n−6n(n−1).