Students Lying to Teachers

It’s sad, but many students lie to teachers and get away with it. That’s why I love to read about stories where teachers catch the students red-handed.

Here’s one story I especially enjoyed:

"My dad heard this story on the radio. At Duke University, two students had received A’s in chemistry all semester. But on the night before the final exam, they were partying in another state and didn’t get back to Duke until it was over. Their excuse to the professor was that they had a flat tire, and they asked if they could take a make-up test. The professor agreed, wrote out a test and sent the two to separate rooms to take it. The first question (on one side of the paper) was worth 5 points, and they answered it easily. Then they flipped the paper over and found the second question, worth 95 points: ‘Which tire was it?’ What was the probability that both students would say the same thing? My dad and I think it's 1/16. Is that right?"

Source: excerpted from Marilyn vos Savant (IQ 228, the highest recorded around the world), Parade Magazine, 3 March 1996, p 14.

Besides being a nice story, the situation raises some interesting questions in game theory.

The answer is no. It's not right. If the students were lying, the correct probability of their choosing the same answer is 1/4.

Again, let's look at the sample space. (response of Student 1, response of Student 2)
{ (RF, RF); (RF, LF); (RF, RR); (RF, LR);
(LF, RF); (LF, LF); (LF, RR); (LF, LR);
(RR, RF); (RR, LF); (RR, RR); (RR, LR);
(LR, RF); (LR, LF); (LR, RR); (LR, LR) }
4 /16 = 1/4

Alternatively, we can look at the situation this way. Student 1 can choose anything. Suppose he chooses RF, then Student 2 has a probability of 1/4 getting RF.

Of course, this is based on the assumption that both students are lying and their choices are independent and importantly, they have equal chances to choose one of the four wheels.

In reality, people prefer RF tire (58%), with the other three tires of similar probability, LF (11%); RR(18%); LR(13%). (Source: Introduction to probability By Charles Miller Grinstead, James Laurie Snell, p.40)