Lockers

Probability Level 3

The grade 12 class of École Sainte-Anne has 100 students and exactly 100 lockers available.

The lockers are stacked as indicated in the picture and the row is exactly 50 lockers wide. (50 x 2 =100)

Romeo and Juliette would like to have lockers that are adjacent but the teacher has decided to randomly assign the lockers.

What is the probability that their lockers will be adjacent?

If the probability can be expressed as a b \dfrac ab for coprime positive integers a a and b b , what is a + b ? a+b?

Note : We define adjacent as having a common edge.


The answer is 2549.

Paul Fournier by Paul Fournier , 70, Canada

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1 solution

Paul Fournier
Jan 30, 2016
  1. Each locker has 3 adjacent lockers with the exception of the first 2 lockers and the last 2 lockers.
  2. The lockers at the ends have 2 adjacent lockers.
  3. We have (100-4)*3+4(2)= 296 ways where the lockers are adjacent.
  4. The lockers can be assigned to Romeo and Juliette in 100*99=9900 ways.
  5. P= 296/9900 = 74/2475 and 74+2475=2549
4 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

The tricky part here is to calculate the number of pairs of adjacent lockers. Proper cases can help make this easier.

E.g. There are 49 horizontal pairs in the top row, 49 horizontal pairs in the bottom row, 50 vertical pairs.

Another way to think about it, as mentioned by Paul, is that the 4 corner lockers will have 2 neighbors, the perimeter lockers will have 3 neighbors and the interior lockers (of which there are none in this particular case) will have 4 neighbors.

My solution is a little bit different: the teacher can assign the lockers in C:100,2( simple combination). In a row of n lockers there are n-1 adjacent lockers( obtained trough drawing for small cases), hence 49*2( rows). We have to add 50 cases in which the lockers are adjacent by up and low edge. The result is the same. Did you think that order is important in this case( romeolocker, juliette locker is different from juliette locker,romeo locker?)

guido barta - 5 years, 4 months ago

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I had to consider ORDER because of the way i counted the adjacent lockers i.e. i counted each pair of adjacent lockers twice. i.e. i considered RJ different from JR and divided by 100 x99 rather than 100x99/2 like you did.

Paul Fournier - 5 years, 4 months ago

Another approach to this is using the conditional probability. What is the probability that Juliette's locker is adjacent to Romeo's, given Romeo has chosen any locker? We know that every locker has a 1/100 probability of being chosen by Romeo. Then the probability that Juliette's locker is adjacent to Romeo's is 2/99 for 4 of the lockers and 3/99 for 96 of the lockers. We can sum them up to get the final answer: (1/100) * (2/99) * 4 + (1/100) * (3/99) * 96 = 74/2475.

Judy Gu - 3 years, 9 months ago

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