In Part 2, we met the central limit theorem: if each sample is made up of a sufficiently large number of draws from a box, then the sample means (or sums) will follow an approximately normal distribution. We also saw how to check whether the number of draws is large enough — repeatedly form samples, calculate the mean (or sum) of each one, and see whether the histogram of those values looks normal.
Why do we care about that shape? Because once we know the sample sums (or means) follow a normal distribution, we can start to answer probability-based questions about them.
For example: what is the probability of flipping 60 or more heads out of 100 coins? We start by writing this as a box model. We put a '1' ticket in the box (representing a head) and a '0' ticket (representing a tail). Flipping 100 coins is the same as drawing 100 times from that box, and the number of heads is simply the sum of the 100 tickets we draw. This box model is pictured to the right.
To turn that box into a probability, we work through four steps. These four steps are exactly the four sections of the Box Model Playground below, in the order they appear:
Don't worry if some of this still feels vague — each of the four sections below explains its own step in much more depth, and updates itself automatically for whatever box you set up. The playground already starts configured for the coin question above, so the best first move is to scroll down and follow the four sections through in order.
The first suggested exercise is to recreate the scenario above, but with a slight change. What is the probability that if you flipped 100 coins, you would
observe 70 or more heads?
To do this:
The answer is 3e-05, which means 0.00003 (0.003%).
Extension Question: What is the chance that you observed 40 to 70 heads (inclusive)?
You have one ice cream left at home, and you and your brother desperately want it. Your brother turns to you and says, "Let's play a little game. I want
you to roll a 6-sided die 50 times. Every time you roll it, write down the face that the die landed on, and at the end, add up the 50 numbers you recorded
rom the 50 rolls. If the sum is greater than 150, then you can have the ice cream, and otherwise, I get to eat it." You agree to his little game.
To do this:
We won't give you the EXACT answer this time, but it should be above 90%.
Extension Question: For the game to be fairer, there should be a 50% change that either you or your brother could
win. Instead of 150, what should the sum be changed to?
Next week, you and your brother are in the same predicament. One ice cream left - who gets it? Your brother proposes a new game. You roll a 6-sided die 50 times and take the sums, just as you did last time. If the sum is between 145 and 175 (inclusive), you get the ice cream. Otherwise, your brother gets it. What is the probability that you get the ice cream?
First we need to create the box. Follow steps 1, 2, and 3 to configure the box to be whatever you wish to model. The box will output on the right according to the settings that you pick.
First, we need to specify the tickets that we will be adding to the box.
In the text box below, enter the tickets that you wish to place into the box. Only numbers can be added, and each ticket should be seperated by a comma. For example, if you want to have one '1' ticket and one '0' tickets in the box, enter 1,0.
Error: One or move values that you added cannot be interpreted. Please carefully check what you entered. You must enter at least 2 valid tickets. Setting contents of the box to 1,0,0,0.
Second, we need to specify the number of draws (with replacement) that we will be taking from the box:
Error: The value for the 'number of draws' must be an integer greater than or equal to 1. Setting value to 25 until the error is resolved.
Third, we need to specify whether we will be representing the sample using the mean or sum of the draws:
Lower Boundary
Upper Boundary