Geometry – Day 62 – Geometric Probability

Come for:

  1. The probability of shapes.

Stay for:

  1. Problems that frustrated the students and, consequently, their teacher.
  2. An abysmal three problems completed in two hours.
  3. A lesson I should’ve skipped.


  1. None


Slide Deck

  1. 99 days. Associated Press
  2. Doubled the lengths for ‘em.
  3. It’s always interesting to see who picks green and who, sensing a bluff, picks blue. The numbers have been chosen for ambiguity. If you miss you throw again.
  4. Define probability: what you want to happen divided by everything that could happen.
  5. Calculate
  6. Calculate
  7. Calculate
  8. This should not have been the frustration it was for all of us. I still don’t know how I could’ve scaffolded this one better.
  9. Point out that they all add up to 100%.
  10. Talk about payoff. Talk about how Vegas pays off less than what is mathematically fair. That’s the house advantage.
  11. Pass out some graph paper. Ask for a bet right here, right now.
  12. Does anyone want to change her bet?
  13. Talk about how you can use Pick’s Thm. here
  14. I’m rounding the payoff odds for the sake of the next screen.
  15. This is as far as we got today. I suck.
  16. Each ring is 4 inches thick. The bonus triples your score.
  17. Have them shade the area that gets them this score.
  18. Have them shade the area that gets them this score. A much more complicated question than the last.
  19. Never got around to this. Probably won’t until next year.

Notes & Revisions:

  1. I’d rather overplan than underplan but that was a lot of wasted planning.
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Hey Dan,

    Cool lesson…. but am I missing something in slide 21? How am I able to calculate the 17% chance of hitting within the bonus area if its dimensions aren’t (or at least, don’t seem to be) defined? Maybe I missed something from an earlier lesson, or maybe there’s some way of playing darts out there in California where those “bonus” wedge shapes are already known.

    And after writing that comment, I must recognize that small, picky questions like that distract from the fact that overall it’s a really good lesson and the graphics still must be keeping you up at that coffee shop until WAY too late each night. I think I’ll hit my 7th graders with the “Place Your Bets” (slide 3) problem after spring break (which we’ve just begun in Orlando, and yet here I am in Syracuse, NY at my brother’s house in sub-freezing temperatures!).

  2. Whoops. No, you didn’t miss anything. I neglected to include the textbook information, which had each bonus sector measuring 30°. That probably makes more sense.

    What doesn’t make sense is spring break in sub-freezing temperatures but I guess it’s hard to argue with some r&r no matter where you are.

  3. What is the probability that they will miss the dart board all together? If I were one throwing, that’s where I’d place my bets.

  4. Thanks for the one missing “clue” – my best guess was that the chord of each of those arcs (created by the intersection of the angular lines and the outermost circle) was equivalent to the radius of the inner circle, but that would have required a fair bit of juggling to end up with the measure of the angles or something like it in radians, etc.

    As for the temps, I can only say that the snowy northeast helps to make me appreciate when I fly back home to 80+ degrees. And the high temperatures are only headed in one direction back home for the next 8 months or so, and that’s UP. Also, spring break in Florida means grossly overcrowded beaches, so that is the LAST place that I want to spend my week off.

  5. What should you have skipped?

    I skip, btw, geometric probability. We don’t have regular probability until far later. We do work with the same figures you propose.

    One idea (not a lesson, just a challenge, and will require skills they may not have acquired yet), rerun slide #8 with lengths, hm, 2, a and b, such that the areas of the three regions are equal (or the probability of hitting each region is equal), (or the probability of hitting each region is in the ratio 1:2:3).

    Ah, the mind wanders.

  6. I should’ve skipped the entire lesson. The rest of the department did. Geometric probability isn’t a blueprinted standard in California and, personally, I’m of the mind that if I’m going to go off the standards reservation, I should have a compelling reason. Usually, I’ll jaunt off in favor of something the class finds stimulating and fun. This lesson was neither a standard nor fun. Whoops.

    I dig that question, though, and my kids have all the prereqs for it. Just gotta find the right place. Thanks.