From pages 278-279, do exercises 25, 28, and 29.
I wanted to post a few thoughts from recent homework assignments, based on students’ questions. If you have questions about homework, please post them here!
My answers to 11 and 12
11a) The problem here is that using a random number 0-9 assumes that every number of heads is equally likely. This isn’t true. In order to get all 9 coin flips to land heads, that’s a .5^9 probability. But in order to get just 4 or 5, there’s a much higher probability of that. The random numbers we generate for a simulation must represent the actual phenomenon being simulated (in this case, a single coin flip), not the net result of a whole trial.
11b) This would not be a problem if there were exactly a 50-50 chance the player makes a shot, but we would presume the player is slightly better than that.
11c) The probability the *first* card drawn from a deck is an Ace is 4/52, which is indeed 1/13. But the probability the *next* card is an ace is only 3/51 (and for that matter the probability of it being any other denomination is 4/51). The probability changes with each card that we draw, yet the odds of getting any one number out of 13 possibilities will always be constant.
12a) This is the same issue as 11a. Getting a 2 is a lot rarer a situation than getting a 7
12b) Generally, we assume the chances of a baby being a boy is 50-50, so this is also the same exact problem as 11a. In this case, we would have to generate 5 numbers, one representing each child. We cannot let one random number represent a whole trial.
12c) This is a similar issue to 11b (and others). Running a simulation in this way assumes that each outcome is equally likely.
What if no trial in my simulation results in a success (see 20)
It’s important to remember that your simulation is just that, a simulation, not reality. Of course it is possible for someone to randomly guess and get all 6 questions right, and it’s not hard to calculate the probability of that directly (0.25^6 = .00024). The proper statement to make should no trial result in a success is something to the effect of “My simulation predicts a less than 5% (1/20) chance of randomly guessing all six questions correctly. My friend’s claim that she did so is doubtful.” Remember the Smelling Parkinson’s example from the start of the year: nobody’s simulation achieved 11 correct guesses, but we didn’t say it was impossible
How do I design a simulation for this…?
Simulations are hard to design sometimes, and it’s difficult to offer advice on designs in general since the specifics are always so different. But my biggest suggestion is this: Imagine what actually doing the action described would look like. The only difference between what is actually going on versus what is happening in your simulation is that you’re using random numbers. So instead of actually checking cereal boxes for sports cards, you’re generating a random number for which card is in each box. Instead of actually taking a 6 question quiz, you’re generating a random number to represent whether she got each question right. Instead of actually throwing free throw baskets, you’re generating a random number to represent whether she makes it in or not. Everything else: how long she throws baskets, how many boxes you check, etc., is exactly the same.