Why You Should Have Risked Guessing in the UPCAT
You slowly open your eyes to the roar of an excited audience and the glare of studio lights. You are onstage, on the set of the legendary Monty Hall game show. Applause erupts around you as Monty Hall himself steps forward.
Before you can compose yourself, he goes up to you and explains the rules of the game.
“Before you lie three doors. Behind two are goats; behind one is a car. Whatever you choose, you get to take home.”
You choose Door 1, but before it opens, Monty Hall reveals a goat behind Door 3.
“Now, you can choose to either stay with Door 1 or switch to Door 2. The choice is yours.”
What do you do? You pause for a moment. “Would there be any difference? It is equally likely that the car is behind either of those doors…”
You settle on sticking with your original choice, Door 1, but just as you are about to confirm your decision, a realization pops into your head.
“Initially, there was a one-third chance that I picked the car, since there were three doors to choose from. That means Doors 2 and 3 have a combined success rate of two-thirds. However, when Monty Hall revealed the goat behind Door 3, the chance of Door 3 being the winning door became 0. Thus, the one-third chance of Door 3 being the right choice had to be transferred to Door 2 to give it a two-thirds chance of success!”
Ding ding ding ding! Your alarm clock sets off, and as you check the date on your phone, reality sets in: it is the morning of the dreaded University of the Philippines College Admissions Test (UPCAT). “Oh no,” you whisper. If only you had started reviewing earlier.
Fast-forward to the end of the exam. The last question is difficult. You have never seen a question like this before. Four choices. A wrong answer incurs a 0.25-point deduction; a blank, 0 points; and a correct answer, one point. What do you do?
You hesitate. At first, you consider leaving it blank. If you guess, you have a one-fourth chance of getting it right with one point, but a three-fourths chance of getting it wrong with -0.25 points. Best not to risk it, right? Wrong. You do the calculations in your head. When you multiply the chances with their respective gains...
Chance of getting it right: 25% x 1 = +0.25
Chance of getting it wrong: 75% x -0.25 = -0.1875
Adding them...
0.25 + -0.1875 = 0.0625
Since the number is positive, every time you blindly guess, you are expected to gain an average of 0.0625 points. This is called expected value. If you leave the question blank, your expected value is 0. Mathematicians and statisticians use this to assess whether one should take chances on something given the risks (three-fourths chance of error, -0.25 points per mistake) and rewards (one-fourth chance of being correct, one point per correct answer). Remember, this probability only applies when you make an unbiased random guess–that is, not looking at the question and answering based on what seems more ‘correct.’
You go back to the choices, since you still want to narrow them down. Suddenly, you remember your dream, while Monty Hall quietly whispers in your head. “Which choice leads you into the Maroon and Gold?” You tell Monty to be quiet and let you focus on your exam. As your initial ‘safety’ choice, you select C. Then, as you begin the process of elimination, you soon figure out that D is wrong. Wait a minute, A as well! Now, it is down to just B and C.
Your mind blanks, but just as panic begins to set in, you remember Monty Hall and the weirdly theoretical dream you had that morning. You realize that you are in the same scenario!
It dawns on you that you have a three-fourths chance of getting the point if you switch! Remember that when guessing between any number of options, 1/(total number of options at the start) will always remain the chance that the guess is correct. When you eliminate the two wrong options, your original one-fourth chance never improves, but the other remaining choice now ‘absorbs’ the probability that used to be shared between the other three choices. Thus, if you disregard bias when selecting an answer, you will have three times a likelier chance to win if you switch to B.
While you leave the exam hall, you search online to see if anyone has had any trouble with it before. In fact, many have! Your original dream is known as the Monty Hall problem, a famous brain teaser that has stumped millions since the ‘70s.
Essentially, the Monty Hall problem highlights how new information reshapes probability. Going back to the dream you had, while your initial choice had a one-third probability of being correct, Monty’s action of revealing a losing door acts as a ‘filter’ that boosts the odds of winning after switching.
You sigh as you are assured you made the right decision—to guess on the UPCAT, a test that will determine your future.
To the students who took (or will be taking) the UPCAT, do not be scared and trust the math. Of course, make sure to study well and review. Probability works best when combined with good preparation.
Disclaimer: There is no guarantee that you will always arrive at the expected value, as it is just the anticipated average outcome of a random variable over many repetitions. Just be sure not to blame me when you are disappointed come April.
Sources:
https://betterexplained.com/articles/understanding-the-monty-hall-problem/
