Grand Challenges

Grand Challenge ONE

PROBLEM:

A well shuffled deck of cards is dealt into four hands with 13 cards each, as in the game of bridge. What is the probability that each of the four hands has one of the four aces?

Notes:

This problem is the first one we tackle in the course, “Quantitative Problem Solving” at Baldwin Wallace University. The course has no math prerequisites. We have theater majors, business majors, history majors, and a variety of others.

The problem is not solved with math formulas, but by thinking, wondering, and exploring. It takes few weeks to finally solve it. The students get a lot of encouragement, coaching, and advice from the professors, but it is the students who figure it out. They figure it out by developing a fundamental understanding of probability. We never discuss or use factorials or formulas. They do what makes sense to them. They eventually say, “Oh, I get it, that makes sense.” When they finally solve it, they are proud of themselves because they had no idea how to start and they eventually figured it out. One student said, “I had no idea how smart I was!” This is understandable. If the teacher doesn’t challenge students to struggle with hard problems, they’ll never have the opportunity to experience success after a long struggle. This so important. This is part of being a human being. Struggling and then succeeding. A good example of this is an infant learning how to walk. It takes months, but they eventually succeed, and they are so proud of themselves.

Coaching:

If a student has not been asked to solve challenging problems without a formula or a series of steps to follow, tackling a problem like this can be overwhelming and frustrating. It helps to have a coach to provide guide-rails, encouragement, and to ask insightful questions to get the student started on the problem. Sometimes, parents, grandparents, and other relatives can be great coaches. Often a teacher can be an excellent coach. If you or someone you know is interesting in developing the ability to think hard to independently reach a new level of understanding and he/she does not have access to a coach, this is a service we offer over the internet. If you are interested coaching (not someone showing you how to get the answer), fill out this form and we’ll get in touch with you to see if there is a good fit.

Grand Challenge TWO

PROBLEM:

Six standard dice are rolled. What is the probability that exactly four of the possible numbers appear? Note that a gambler in the 1700s knew that the result was very close to 50%, but he nor any mathematician he asked could calculate it. Can you?

Grand Challenge THREE

PROBLEM:

Imagine a stock that has a value of $100 on January 1st. On the 15th of each month a fair coin is tossed, and the value of the stock is increased by $1 if the coin is heads and the value of the stock is decreased by $1 if the coin is tails. Two investors, Alice and Bea, each buy one share at the beginning of the year. Bea would like to limit her loss to $5 so she applies a stop-loss order at $95. If the stock price drops to $95 anytime during the year, she sells her share for $95 and accepts the $5 loss. Which investor, if either, has a higher expectation at the end of the year?

Grand Challenge FOUR

PROBLEM:

Your favorite uncle just had a successful trip to the casino and offers you the chance at a $100 casino chip. He starts by explaining that there are three identical bags and six casino chips. One bag contains two $100 chips, one bag contains both a $1 chip and a $100 chip, and the third bag contains two $1 chips.

He then mixes up the bags, so you don’t know which bags contain which chips. Your uncle is interested in developing your probabilistic reasoning, so he offers you the chance to select a sample chip before choosing the chip that you get to keep.

The chip that is sampled is not returned to the bag, so if you choose to select a sample chip, the chip that you get to keep will be one of the five remaining chips.

After sampling a chip, you are free to choose the chip you get to keep from any of the three bags. Of course, your uncle will not allow you to look into the bags to see what is inside and you must declare whether you are opting to select a sample chip before removing the first chip and seeing its value.

Do you sample? If you sample, what are you hoping for? What is the optimal strategy for maximizing the probability that you walk away with a $100 chip?