Discrrete mathematics for computer science random warmup

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