## Thursday, May 18, 2017

### 5/16 Expected Values

My planning process this week went something like this: after last week's talk I either wanted to do some group white-boarding or find a new game to explore. I also was thinking more about combinatorics. I've never done anything on combinations (n choose m) and I mulled choosing that as a theme. Then in the middle of the week the Math Counts finals occurred. Watching the live stream was fun for me and I thought the kids would like that too. So initially, I thought about showing pieces of the video and then pausing and have everyone do the problems on the whiteboard. But after some more thought, I worried that it would emphasize the speed of the competitors too much and I also wanted to dig into the Chicken problem more deeply.

Finally this was the structure I ended up with:

We warmed up with some individual skyscraper puzzles from https://www.brainbashers.com/skyscrapers.asp?error=Y. I really like doing these and they went over well engaging everyone. I cut this short after everyone had at least finished the first one of the set I provided in the interest of time.

After watching the videos http://www.espn.com/watch/?id=3034800&_slug_=2017-raytheon-mathcounts-national-competition I transcribed a few of the problems I liked and thought would be good to try on the board:

#### Whiteboard Problems from the video:

• Caroline is going to flip 10 fair coins. If she flips  heads, she will be paid \$. What is the expected value of her payout?

• Sammy is lost and starts to wander aimlessly. Each minute, he walks one meter forward with probability 1/2 , stays where he is with probability 1/3 , and walks one meter backward with probability 1/6. After one hour, what is the expected value for the forward distance (in meters) that Sammy has traveled?

• A finite geometric sequence of real numbers with more than 5 terms has 1 as both its first and last terms. If the common multiplier is not 1 what is the value of the 4th term?

• The length of a 45 degree arc on circle p, has the same length as a 60 degree arc on circle q. What is the ratio of the areas of circle p to circle q?

• The novel Cat Lawyer is 300 pages long and averages 240 wd/pg. The sequel Probably Clause is 60 pages longer and 30 more words per page. Probable Clause has what % more words?

• Ian is going up a flight of stairs. Each time he takes 1,2 or 3 steps. What is the probability that he steps foot on step 4?

• How many 6 digit integers are divisible by 1000 but not 400?

But I decided to focus on the final question:

"In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of un-pecked chicks?"

To build up to it, I started with discussing expected value and used dice questions as starters in a group.

1 What's the expected value of a single roll of a six sided die?
2. What's the expected value of 2 rolls?

After we went over the concepts as a group, out came the blue markers and  I had everyone work on the followup problems up on the various whiteboards.

3. What's the expected value of the product of 2 rolls?
4. What the expected value of the product plus the sum of the rolls?