Notes for February 25 and 27

The dependent probabilities in a 52 card deck

One of the simplest mathematical models of dependency is sampling without replacement, which is the way most card games or lotteries or the game of Bingo works. You have a set of outcomes which get effectively randomized and a trial is performed, meaning a card is taken from the deck or a ping pong ball is removed from the hopper or a bingo marker is removed from the spinner. Once removed, the number of possible outcomes has been reduced by one and probabilities for success and failure of certain outcomes change.

Looking for an ace: There are 52 cards is a standard deck and 4 of them are aces. If I draw a card from a randomized deck, the chances are 4/52 = 1/13 ~= 7.7% that the card will be an ace. What are the chances the second card is an ace?

That depends on the first card.

Probability that the second card is an ace, given the first card is an ace is 3/51 = 1/17 ~= 5.9%.

Probability that the second card is an ace, given the first card is not an ace is 4/51 ~= 7.8%.

Unlike the mathematical model of free throw shooting where we re-calculate the probabilities by adding the most recent make or miss into the percentage, which means a miss brings the odds down and a make brings the odds up, not getting an ace makes the odds a little better next time, and getting an ace makes the odds worse.


This is the formula for the dependent probability model of sampling without replacement is given at the left. The two numbers in parentheses are a binomial coefficient, the numbers you get when you use nCr on your calculator, which I pronounce "n choose r" in class. The pairs of numbers that look like a base and an exponent, except that the exponent is underlined, is the convention developed by Donald Knuth at Stanford for writing the numbers that you get on your calculator using the nPr function, which I pronounce "n fall r", referring to the name "the falling factorial". If we think about a deck of cards, the lowercase letters refer to the size of the hand n, where r is the number of successful trials (r for right) and w is the number of unsuccessful trials (w for wrong), and r+w=n. The uppercase letters refer to the size of the deck, where T is the size of the deck, G is the number of cards we consider success if we draw them and B is the number of cards we consider a failed trial if we draw them. The letter T stands for Total, G for Good and B for Bad. Again, we have an equation, G+B=T.

Example: If we want consider drawing a heart a success and anything else a failure, what is the probability of drawing three hearts and two non hearts in a five card hand from a well-shuffled 52 card deck.

Here are the six numbers we need.
n = 5
r = 3
w = 2
T = 52
G = 13
B = 39

On a TI-30XIIs, here are the keys you would press.

5[prb][right]3×13[prb]3×39[prb]2÷52[prb]5[enter]

The calculator will read as follows.

5 nCr 3*13 nPr 3*39 nPr 2/52 nPr 5
0.081542617

This means the probability of exactly three hearts and two cards of some other suit is about 8.15%.

Let's say instead the deck had 10,000 cards and 2,500 hearts. our numbers would change.
n = 5
r = 3
w = 2
T = 10000
G = 2500
B = 7500

On a TI-30XIIs, here are the keys you would press.

5[prb][right]3×2500[prb]3×7500[prb]2÷10000[prb]5[enter]

The calculator will read as follows.

5 nCr 3*2500 nPr 3*7500 nPr 2/10000 nPr 5
0.0878613102

The difference is small, but the second number is much closer to the odds of 3 out of 5 when the probability of success is .25 every time


5 nCr 3*.25^3*.75^2
0.087890625

The point of this is that as the size of the deck gets larger, dependent and independent probabilities get closer together.

When we use categorical data, the most important statistic we try to predict is the proportion of a value in the population, which we call p, which we will estimate using the proportion from the sample, known as p-hat.

Again we will create a confidence interval, and the formula for standard deviation is very different.

sp-hat = sqrt(p-hat * q-hat/n)

The confidence level multipliers for xx% are taken from the z-score table (Table A-2) instead of the t-score table, and this is because the standard deviation for the sample and the standard deviation for the population are expected to be relatively close to one another. The values for the CLMxx% are given on the first page of your yellow sheets in the lower left hand corner.

CLM90% = 1.645
CLM95% = 1.96
CLM99% = 2.575

Example: Consider data sets #1 and #2, and the proportion of males. Let's find the 95% confidence interval for the underlying population, which we will limit to students at Laney who take statistics.

Data set #1:
n = 38
f(males) = 18
p-hat(males) = 18/38 ~= .474
q-hat(males) = 1 - p-hat(males) ~= .526

.474 - 1.96*sqrt(.474*.526/38) < p < .474 + 1.96*sqrt(.474*.526/38)
.315 < p < .633

Given this sample of 38 students, we are 95% confident the percentage of male students taking statistics at Laney is between 31.5% and 63.3%.


Data set #2: n = 42
f(males) = 12
p-hat(males) = 12/42 ~= .286
q-hat(males) = 1 - p-hat(males) ~= .714

.286 - 1.96*sqrt(.286*.714/42) < p < .286 + 1.96*sqrt(.286*.714/42)
.149 < p < .423

Given this sample of 42 students, we are 95% confident the percentage of male students taking statistics at Laney is between 14.9% and 42.3%.

Data sets #1 and #2 combined: n = 80
f(males) = 30
p-hat(males) = 30/80 = .375
q-hat(males) = 1 - p-hat(males) = .625

.375 - 1.96*sqrt(.375*.625/80) < p < .375 + 1.96*sqrt(.375*.625/38)
.269 < p < .481


Given this sample of 40 students, we are 95% confident the percentage of male students taking statistics at Laney is between 26.9% and 48.1%.

Notice how much our intervals disagree with one another. This is because our best point estimates from the three sets are .474, .286 and .375. Also notice that the width of the 95% confidence interval tends to be smaller as n gets bigger. When the sample size is 38, the width of the confidence interval is .318. At n = 42, it is .274 wide. At n = 80, the width is .212. The most common way to make a confidence interval narrower is to increase the size of the sample.

There are two other ways to change the width. If you ask for a higher confidence level, the interval will get wider. If p and q are close to 50%, the confidence interval will be wider than if the are both far away from 50%.

Class notes for 5/11: p-values and Confidence Levels

The test statistics we will use for hypothesis testing will differ, sometimes using z-scores, other times t-scores and yet other times the chi-square table. What all these tests have in common is that they correspond to probabilities, called p-values.

The idea is that if we have a test with 90% confidence, we will only reject H0 if an event happens 10% of the time or less.

Likewise, 95% confidence means we reject H0 when the probability of the event is less than or equal to 5%.

99% confidence means we reject H0 with tests that happen 1% of the time or less.



This table shows what p-value corresponds to the threshold where we reject the null hypothesis, which technically is the same as accepting the alternate hypothesis. When it's one tailed high, the p-value threshold is easy, .9 for 90% confidence, .95 for 95% confidence and .99 for 99% confidence. For one tailed low, the pattern is that the p-value equals 100% minus the confidence level. For the two tailed test, the two tails have to add up to 100% minus the confidence level, so the 90% confidence level threshold is at the p-values of 5% (.05) and 95% (.95).

Remember, just because a test convinced us to reject the null hypothesis doesn't mean the null hypothesis is false. It could still be true, and we would be making a Type I error.

If we use the 90% confidence level, we should make Type I errors about 10% of the time.

At the 95% confidence level, the probability of Type I errors is 5%.

At 99% confidence level, the probability of Type I errors is 1%.

The lower likelihood of errors is why most tests done on medical data is done at the 99% confidence level.

The p-value is often published to show just how well the test did. Maybe you only asked to prove something to 90% confidence on a one-tailed high test, but the p-value is .9978. This shows people that read the findings that it would be strong enough data to reject H0 at even higher confidence levels.