Lab 10

Why?

Why do we need to concern ourselves with probability? Because we're now moving towards talking about inferential statistics, that is making claims about populations based on information from samples. In other words, because we're using samples instead of testing every member of a population, we're moving away from the realm of certainty and into the realm of making our best educated guess (with a willingness to be wrong sometimes). This is the realm of probability.

So we'll start the lab by talking about basic probability theory and then we'll begin to apply it to statistical reasoning.

For much of the course we've talked about sampling error and variability. We've further broken these things down into random and biased (non-random) things. We've talked about ways to try to reduce bias.

This lab focuses on how we deal with randomness.

For example, if randomly select individuals from the distribution of heights, we will get a variety of heights. However, if we did this a lot, then we'd see that in the long run, most of the heights will be right around the mean of the heights for the population. In other words, while we don't know from person to person what height we'll pick, as a group we can predict what most of the heights will be.

Random (chance) behavior is unpredictable in the short term, but has a regular and predictable pattern in the long run. The probability of any outcome of a random phenomenon is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions.

So it is critical to differentiate the "short term" from the "long term" when thinking about randomness.

Download and open this Excel file. This file shows the results of N virtual coin tosses. That is, it simulates what would happen if you tossed a series of coins. The number tosses can be controlled by changing the box next to "N" or sample size. With real coins, the probability of getting head (or tails) is .5. In the simulated coin tosses, you can also control the probability of getting heads on each toss. Hitting the F9 key generates new data.

Blackboard 1) If you set N to 10 and the probability to 0.5 and repeatedly hit F9 (20+ times) on your keyboard, what happens to the proportions of heads and tails?
Blackboard 2) If you set N to 100 and the probability to 0.5 and repeatedly hit F9 (20+ times) on your keyboard, what happens to the proportions of heads and tails?
Blackboard 3) If you set N to 1000 and the probability to 0.5 and repeatedly hit F9 (20+ times) on your keyboard, what happens to the proportions of heads and tails?

You should notice that there is a relationship between N and the variability of the proportions of heads and tails and you hit F9 repeatedly (i.e., as you resample from the population of coin tosses).

Basic probability

We deal with probabilities everyday.
- lotto tickets, weather forcasts, medical reports on the news (e.g., risks of cancer)

In a situation where several different outcomes are possible, we define the probability for any particular outcome as a fraction or proportion. If the possible outcomes are identified as A, B, C, D, and so on, then:

	Probability of A = number of outcomes classified as A
	   		   total number of possible outcomes
                

The total number of possible outcomes is called the sample space S

Some Rules of probability

• Any probability is a number between 0 and 1.
• All possible outcomes together must have a probability equal to 1.0
• The probability that an event does not occur is 1 minus the probability that it does occur.
• If two events have no common outcomes, then the probability that one or the other occurs is the sum of their individual probabililties.

Consider a more concrete example:

you are playing War (the card game) with your kid sister, each of you has your own deck of 52 cards. She picks the Queen of hearts from her deck. What are the odds that you'll pick the Queen of hearts from your deck?

There are 52 different cards in a deck, so the sample space is 52. There is only one queen of hearts. So:

	prob of Q-hearts =   ____picking the Queen of hearts ___
			    total number of possible cards picked

		         =  1 / 52

Notationally we can express this probability as: p(Queen hearts) = f / N = .019


Facts about standard decks of cards: There are four suits (clubs, diamonds, hearts, and spades). Clubs and spades are black and hearts and diamonds are red. Each suit has 13 cards (2 through 10, Jack, Queen, King, and Ace). Thus, there are four 2's, four, 3's, and so forth all the way to four Aces.

Blackboard 4) What is the probability of selecting a red card from a standard deck of playing cards?
Blackboard 5) What is the probability of selecting a club from a standard deck of playing cards?
Blackboard 6) What is the probability of selecting a King from a standard deck of playing cards?
Blackboard 7) What is the probability of selecting a face card (Jack, Queen, or King) from a standard deck of playing cards?
Also answer 2 additional Review Questions on Blackboard (questions 8 and 9).