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For the probability problems below, please select one question to work on, share your computations, and provide your answer with an explanation.
The Birthday Problem – There are 23 people in this class. What is the probability that at least 2 of the people in the class share the same birthday?
The Game Show Paradox – Let’s say you are a contestant on a game show. The host of the show presents you with a choice of three doors, which we will call doors 1, 2, and 3. You do not know what is behind each door, but you do know that behind two of the doors are beat up 1987 Hyundai Excels, and behind one of the doors is a brand new Cadillac Escalade. The cars were placed randomly behind the doors before the show, and the host knows which car is where. The way the game is played out is as follows. The host lets you choose a door. Assume you choose door #1. Before he opens door #1 to let you see what you have chosen, he opens one of the remaining doors, say door #3, to reveal a Hyundai Excel (he will always open one of the remaining doors that has the booby prize), and asks you whether or not you want to change your choice to door #2. What do you tell him?
Flipping Coins – If you flip a coin 3 times, the probability of getting any sequence is identical (1/8).
There are 8 possible sequences: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
Let’s make this situation a little more interesting. Suppose two players are playing each other. Each player chooses a sequence, and then they start flipping a coin until they get one of the two sequences.
We have a long sequence that looks something like this: HHTTHTTHTHTTHHTHT…. We continue until one of the two wins.
Do you think this is a fair game, and that under these rules each sequence has an equal chance to appear first?
Think again! If you chose HHH and I chose THH, I have a much higher chance that you do!
The only way that you win is if the first three tosses are HHH. In any other event, I win.
Agree? Do you see why?
For the sequence HHH to appear anywhere except the first three flips, it must come after a T, right? So, the actual sequence for you to win is THHH.
But if there is a sequence of THHH then I already won before that sequence is over (because my sequence was THH).
So, THH will win 7 times out of 8. HHH will only win if the first three are HHH (a one in eight chance).
Suppose you are going to flip a coin until you get the sequence HTH. Say this takes you x flips. Then, suppose you are going to flip the coin until you get the sequence HTT. Say this takes you z flips. On average, how will x compare to z? Will it be bigger, smaller, or equal?
Disease Testing and False Positives – Assume that the test for some disease is 99% accurate. If somebody tests positive for that disease, is there a 99% chance that they have the disease?
A Girl Named Florida – Here’s a three-part puzzler:
Your friend has two children. What is the probability that both are girls?
Your friend has two children. You know for a fact that at least one of them is a girl. What is the probability that the other one is a girl?
Your friend has two children. One is a girl named Florida. What is the probability that the other child is a girl?
The Value of Variance – More often than not, when we are presented with statistics we are given only a measure of central tendency (such as a mean). However, lots of useful information can be gleaned about a dataset if we examine the variance, skew, and the kurtosis of the data as well. Choose a statistic that recently came across your desk where you were just given a mean. If you can’t think of one, come up with an example you might encounter in your life. How would knowing the variance, the skew, and/or the kurtosis of the data give you a better idea of the data? What could you do with that information?
Example: Say you are an executive in an automobile manufacturer, and you are told that, for a particular model of new car that you sell, buyers have on average 2.2 warranty claims over the first three years of owning the car. What would additional information on the shape of your data tell you? If the variance was low, you’d know that just about every car had 2 or 3 warranty claims, while if it was high you’d know that you have a lot of cars with no warranty claims and a lot with more than 2.2. The skew would provide similar information; with a high level of right skew, you’d know that the average is being brought up by a few lemons; with left skew you’d know that very few of the cars have no warranty claims. The kurtosis (thickness of the tails) would help you get an idea as to just how prevalent the lemon problem is. If you have high kurtosis, it means you have a whole bunch of lemons and a whole bunch of perfect cars. If you have low kurtosis, it means that you have few lemons but few perfect cars.
Select and discuss one of the following probability rules:
In 6-sentences or more, explain how the rule applies at the workplace or personal life experience.
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