WT - Disguised Combinatorics (Advanced)

[krishna626]

Can someone please explain the example given on page 76 of the Word Translations strategy guide (3rd edition)?

I do not understand why the answer is not 1/4. Isn't the question asking what is the probability that Alicia walks south for two blocks before going east? Alicia has two options (south or east) and has to make that choice twice. 1/2 * 1/2 = 1/4.

I don't understand why it matters if she then needs to walk east and then south for 3 blocks or 10 blocks. Those other choices are independent if she chooses to walk south twice from the beginning.

Thanks for any help you can provide.

Krishna

[Ben Ku]

Hi Krishna,

In this question, there are a few constraints:
(1) Alicia can only walk on the grid (so she cannot walk south 10 blocks).
(2) She has to start from the Home and end at the School.
(3) She can only travel south or east one block at each intersection.

The probability of is (desired outcomes)/(all possible outcomes). In this question, the desired outcomes are all the routes she can take from Home to School given the first two are going south.

As described on page 77, "all possible outcomes" means she must travel south three times and east three times, so we want all anagrams of SSSEEE, which is 6!/(3!3!) or 20.

For "desired outcomes," she MUST travel south for her first two steps, so we want all outcomes SSXXXX. Among the X's, one must be S and the other three E's, since we have already traveled S twice. In other words, we need all to find all the possible combinations of XXXX which are the anagrams of SEEE: 4!/3! or 4.

Your approach of the problem answers the question: if Alicia were to walk two blocks, what is the chance that she will walk S twice? In this case, she has a 1/2 chance of walking S her first block, and 1/2 chance of walking S her second block. It does not answer what is the chance of going from home to school.

I hope that makes sense.