A friend was telling me that he knew a college professor who would split his class in two, and have one team write a list of numbers randomly generated by a computer on the blackboard while the other team tried to make up numbers that looked randomly generated. Apparently, this professor would walk into the room and immediately point to the one that was truly randomly generated, not created by people.
Stories and brainteasers about probability, statistics, and mathematics in general really make math concepts stick with students. Any favorites that you'd like to share?
Christine
Community Manager
I was hoping to see some great brainteasers out here since I am teaching a Thinking in Data and Probability course and would love to share these with the participants. So far I see none. I am teaching the second day of the course on Nov. 30 so I hope to see some great brain teasers out here before then. Thanks ahead of time.
This month's brainteaser from Illuminations (answer will be posted next Monday):
In a four‐digit number, the sum of the digits is 10. All the digits are different. What is the largest such four‐digit number?
I'll guess 7210.
At first I thought 9100—then I went back and read the directions a bit better and saw that all digits had to be different. That eliminates 9100 (and 9010, 9001), 8200 and 8110 etc, and 7300. Next highest (I think) would be 7210.
And the answer is....
Christine
I have two problems for you, Marcia.
1) Counterintuitive Cards -- You have 7 cards: two are red on both sides, two are black on both sides, and three are red/black. You choose a card at random, and one of its sides is red. What is the probability that the other side is also red? (Hint: The answer is not 2/5, though that's what everyone thinks.)
2) Filling a Bag -- You place some red and some blue marbles in a bag. The probability of drawing two marbles of the same color is 1/2. How many marbles of each color are in the bag? (Hint: The correct answer does NOT have the same number of red and blue marbles.) Note that there are many possible answers. Try to find one. Then try to see if you can find others and identify a pattern.
Thanks so much and in time for the class!! I will use these with the participants.
Here's my favorite:
You're on a game show, facing three closed doors. Behind two of the doors is nothing; behind the third is a new car. You're allowed to pick only one door, so choose door A. It remains closed while the suspense builds.
The game show host, who already knows where the car is, decides to open one of the doors with nothing behind it first. There happens to be nothing behind door B, so he opens it. There now only two closed doors, A and C.
He then presents you with this challenge: do you want to change your answer to door C, or stick with your original choice of door A? Is it any advantage to switch?
Christine
Christine
Thanks for reminding me of this one. I had this during one of my data classes I took. Its a good one!
My understanding is that it's better to switch, because the game show host has a 2/3 chance of winning at the outset, and that doesn't change when he shows you the door with nothing behind it (because we always knew that there would be nothing behind at least one of his doors.)
Although—someone once argued that the game show host needs to *know* that he is opening the door with nothing behind it for this to remain true. In other words, if the game show host simply opened one of his two doors without knowing whether there was nothing or a car behind it, and there was nothing, that changes the probability and you're now on equal footing with a 50/50 chance.
Think that's the case? I truly don't know—I'll let someone who is better with probability weigh in on that one!
Christine
And one more (although this is more of a logic question):
Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person?
A) Yes
B) No
C) Cannot be determined
Another good one. What if I don't know the answer.....
Here's the answer as it was written up (in a viral email):
More than 80 percent of people choose C. But the correct answer is A. Here is how to think it through logically:
Anne is the only person whose marital status is unknown. You need to consider both possibilities, either married or unmarried, to determine whether you have enough information to draw a conclusion. If Anne is married, the answer is A: she would be the married person who is looking at an unmarried person (George). If Anne is not married, the answer is still A: in this case, Jack is the married person, and he is looking at Anne, the unmarried person.
This thought process is called fully disjunctive reasoning—reasoning that considers all possibilities. The fact that the problem does not reveal whether Anne is or is not married suggests to people that they do not have enough information, and they make the easiest inference (C) without thinking through all the possibilities.
(By the way...I was one of those people who didn't think through all the possibilities...!)
Christine
Christine -- The professor was using the idea of "clusters" when identifying which set was real and which one was fake. Turns out, sets of data that are truly random usually contain clusters that most people wouldn't expect. For instance, if you flip a coin 100 times, it's actually quite likely that at some point you'll get 7-8 heads or tails in a row. But if you ask people to write a set of 100 H's and T's that could represent 100 coin flips, most people wouldn't put in a string of that many consecutive of the same type. (I generated a string of H's and T's in Excel; notice that it has a cluster of 9 T's, two clusters of 4 T's, and one cluster of 4 H's.)
T, T, T, H, T, T, H, H, T, H, T, T, T, T, T, T, T, T, T, H, T, T, T, T, H, T, T, H, H, H, H, T, T, T, T, H, T, H, H, H, T, H, T, T, T
Your description is a variation on this, but it works the same way. A truly random set of random numbers would contain similar clusters. In fact, I just generated a set of 186 random numbers (1-10) with Excel, and it has a lot of clusters -- see below. There are 13 pairs of repeating numbers, and there are even 2 triples of repeating digits. Most people who are "faking the data" wouldn't include so many clusters of repeats.
10, 5, 9, 7, 2, 3, 2, 10, 2, 3, 2, 1, 5, 1, 6, 10, 9, 3, 2, 4, 2, 7, 1, 2, 6, 1, 8, 10, 5, 2, 8, 5, 4, 6, 5, 8, 5, 5, 5, 10, 9, 1, 5, 6, 1, 6, 1, 1, 8, 10, 9, 7, 10, 3, 4, 7, 9, 9, 5, 7, 8, 10, 9, 1, 3, 2, 9, 7, 6, 1, 4, 2, 2, 10, 3, 4, 8, 2, 10, 1, 2, 2, 3, 4, 2, 6, 4, 9, 6, 4, 4, 3, 7, 9, 10, 1, 1, 9, 4, 7, 7, 10, 2, 10, 7, 7, 1, 6, 7, 10, 8, 10, 2, 3, 3, 9, 4, 4, 2, 9, 3, 4, 9, 1, 1, 5, 1, 3, 3, 4, 10, 9, 7, 10, 3, 3, 8, 7, 1, 6, 4, 2, 6, 2, 5, 2, 8, 9, 5, 1, 4, 9, 3, 4, 1, 4, 5, 7, 2, 1, 8, 2, 2, 2, 7, 2, 6, 5, 8, 2, 3, 1, 2, 7, 1, 5, 9, 2, 10, 10, 8, 6, 5, 1, 9, 3
That's exactly the answer my friend related...people generating random numbers don't use enough clusters.
But it surprises me to hear that you're likely to get 7-8 heads in a row if you flip a coin 100 times! I had no idea.
Christine