Interesting Probability Problems

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InterestingProbabilityProblems

• Presented by
• Joe Hill
• Director of Math and Technology
• Rockingham County Public Schools

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Problem 1 Choose a number

• 1 2 3 4

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Solution 1

• Approximately 80 of people choose 3. Why?
  Perhaps its our desire to be upper middle
  class or not at the extremes. Mathematics
  cant explain why, but experimentation will
  reinforce the fact that the probability that 3 is
  chosen is much higher than the probability of
  another number being picked.

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Problem 2 Choose a number that no one else
will pick

• Pick a whole number from 1 through 30 which no
  one else in this room will choose. Can you all
  do it?

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Solution 2 (continued)

• Here are the probabilities that a match will
  occur in this room.

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Problem 2 Choose a number that no one else
will pick

• Next pick a whole number from 1 through 50 which
  no one else in this room will choose. Easier to
  do, huh?

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Solution 2 (continued)

• Here are the probabilities that a match will
  occur in this room.

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Problem 2 Choose a number that no one else
will pick

• Finally pick a whole number from 1 through 100
  which no one else in this room will choose.
  Should be a snap to do, huh?

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Solution 2 (continued)

• Here are the probabilities that a match will
  occur in this room.

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Solution 2

• To see how this works, consider four people who
  call their numbers aloud. The probability of
  each NOT having a match with that called out is
• For person 1, P(not match) 1
• For person 2, P(not match) 99/100
• For person 3, P(not match) 98/100
• For person 4, P(not match) 97/100
• For ALL to occur, P(not match)
• 1 x (99/100) x (98/100) x (97/100) 0.941
• So P(match) 1 - P(not match) 1 - 0.941 .059

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Solution 2

• Next, consider ten people who call their numbers
  aloud. The probability of each NOT having a
  match with that called out is
• For person 1, P(not match) 1
• For person 2, P(not match) 99/100
• For person 3, P(not match) 98/100
• For person 10, P(not match) 91/100
• For ALL to occur, P(not match)
• 1 x (99/100) x (98/100) x x (91/100) 0.628
• So P(match) 1 - P(not match) 1 - 0.628 .372

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Solution 2

• Next, consider 24 people who call their numbers
  aloud. The probability of each NOT having a
  match with that called out is
• For person 1, P(not match) 1
• For person 2, P(not match) 99/100
• For person 10, P(not match) 91/100
• For person 24, P(not match) 77/100
• For ALL to occur, P(not match)
• 1 x (99/100) x (98/100) x x (77/100) 0.049
• So P(match) 1 - P(not match) 1 - 0.049 .951

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Problem 3 Guess at SOL questions

• What are you chances of guessing and doing well
  on a Virginia SOL test?
• Well simulate thison each of the next five
  slides is a question followed by four multiple
  choice answers. Guess and see how well you do.

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1. What was the nickname of Bramwell, WV high
schools athletic teams?

• The Aristocrats
• The Millionaires
• The Coal Miners
• The Mountain Lions

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Heres your probability of success
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2. The music for the hymns Dear Lord and Father
of Mankind and Beneath the Cross of Jesus was
written by

• Frederick Maker
• Robert Lowry
• Lowell Mason
• William Doane

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Heres your probability of success
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3. The length of a tennis court from one end of
the inbounds line to the other is

• 72 feet
• 78 feet
• 84 feet
• 90 feet

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Heres your probability of success
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4. The first four digits of the square root of 3
are the same as the birth year of

• Abraham Lincoln
• George Washington
• John Adams
• Thomas Jefferson

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Heres your probability of success
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5. The Virginia Creeper is a bike trail which
goes from Abingdon VA to

• Galax VA
• Bristol TN
• White Top VA
• Harper NC

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Heres your probability of success
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Heres how this works

• For 5 questions, the probabilty of getting
• 0 correct is 0.75 x 0.75 x 0.75 x 0.75 x 0.75
• 1 correct is 5 x 0.25 x 0.75 x 0.75 x 0.75 x 0.75
• 2 correct is 10 x 0.25 x 0.25 x 0.75 x 0.75 x
  0.75
• 3 correct is 10 x 0.25 x 0.25 x 0.25 x 0.75 x
  0.75
• 4 correct is 5 x 0.25 x 0.25 x 0.25 x 0.25 x 0.75
• 5 correct is 0.25 x 0.25 x 0.25 x 0.25 x 0.25

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Which is more likely to give a 50-50 split in
results?

• Spin a new penny, count heads vs tails
• Toss a thumbtack, count side landings versus back
  landings

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Other interesting problems

• If you deal a deck of cards and call out Ace,
  Two, Three, etc. what is the chance that you
  get through the deck without matching the card
  with the word called out?
• If you put letters in a hat and randomly picked
  them out, what is the chance that they would
  spell a certain word?
• Should you switch or stay when playing
  Lets Make a Deal?