Discrrete mathematics for computer science random warmup

Finding a Short Story• Suppose a book of short stories has both shorter short stories and longer short stories.. • Their order within the book is random.. • You want to pick a random st

Random Warmup

Finding a Short Story

• Suppose a book of short stories has both shorter short stories and longer short stories.

• Their order within the book is

random

• You want to pick a random story to read, but you would prefer it is a

shorter one.

Finding a Short Story

• Two Strategies

– Strategy #1: Flip to a random page, back

up to the beginning of that story, and start reading

– Strategy #2: Flip to a random page, go

forward to the beginning of the next story, and start reading

• (Go to first story if you are at the end of the book)

• Does it matter which strategy you use?

To What Kind of Story Is a Random

Page Likely to Belong

• To be concrete, suppose there are 20 stories

– 10 of them are 10 pages long

– 10 of them are 20 pages lon

• So book is 300 pages long

• The Expected #pages in the story on a

random page is (200∙20+100∙10)/300 =

16.7

• When you pick a random page, it will belong

to a longer story with probability 2/3

The Second Strategy Works Better!

• Strategy #1 will have you reading a

longer story with probability 2/3 and a shorter story with probability 1/3

• Strategy #2 will take you to the

beginning of a random STORY

• So Strategy #2 will take you to the

beginning of a longer story with

probability ½ and a shorter story with probability ½

FINIS