Example: Birthdays : In a set of 50 randomly chosen people, what is the probability that any two will have the same birthday?

• Birthdays of the Fall 2018 EMBA class. We have 31 people (including me) and four matches

The result may seem paradixocal; so, here's a link that explains the birthday problem.

Here, I explain this problem for the case of finding two matching birthdays as days of the week (M, T, W, Th, F, S, Su).

Example: The Monty Hall Problem and Monty's show "Let's Make a Deal".  [Probably optional in 2021]

Monty asks: "Do you want to switch the door?"

Here are some explanations of this problem.

• Ron Clarke's explanation
• Numb3rs Math
• And here's a simulation page for this problem that works with Internet Explorer, only.
• Here's my solution using a probability tree.

♦ We will talk about two important concepts (independence of events and mutually exclusive events) before the next Example.

Independent events

I am holding a fair coin (two sides) and a fair die (six sides). For the coin, the probabiilty of getting any side is 1/2. For the die it is 1/6.

Scenario 1: I roll the die and I get a 4. Now I will flip the coin. What is the probability of getting a Heads? It is still Pr(H)  = 1/2. The result of the die roll does not affect the coin's outcome.

Scenario 2; Now I flip the coin and get a Heads (H). I will roll the die now. What is the probability of getting a 4? It is still Pr(4) = 1/6. The result of the coin toss does not affect the die's outcome.

So, here coin toss and die roll outcomes are independent.

Now I roll the die and toss the coin at the same time. What is the probabiltiy of getting a 4 and H? There are a total of 12 outcomes, so it must be 1/12.

We can do it more easily: Pr(4 and H) = Pr(4) x Pr(H) = (1/6) x (1/2) = 1/12. So, for independent events, probability of their joint outcome is simply the product of the individual probabilities.

We note that the 12 outcomes {(1,H), (2,H), ...,(6,H),(1,T), (2,T), ...,(6,T)} are mutually exclusive and collectively exhaustive; i.e., they are all different and they cover all possibilities.

Example: One final and important example: Psy's Gangnam Style on my iPod which has about 500 songs. (I wasted so much money on these amd made iTunes very rich.) See notes.

Two important questions:

(1) What is the probability of hitting Psy's song if I shuffle once?

(2) What is the probability of hitting Psy's song at least once if I shuffle three times?

• Would you believe it? Even the Mac president did the Gangnam  Style dance (see 1:40).

Top of the Page

(c) Topic 3: Random Variables

c.1 Discrete random variables

Example: A gamble based on a coin toss. Fair gamble vs. Unfair gamble. What is the "average" gain in each gamble? Did you just discover the formula for the expected (mean) gain in these gambles?

Example: Pierik's bikes. We will calculate the expected (mean) value of demand, E(X); and variance of demand Var(X).

Here's the Excel spreadsheet  for calculating these quantities for the bicycle shop data.

• R documentation for binomial distribution.
• Background material for binomial probabilities. Wikipedia.

Exercise: Here is a more challenging problem from healthcare area involving the testing of a new drug. Find the solution using Rcmdr. (Answer: 0.74)

c.3 Expected value and variance

Example: Roulette is a board game with a large "house edge." 

Here is the board for American roulette:

The roulette wheel looks like this:

Payout amounts if you win your bet in roulette.

This link simulates the roulette game. Let's try it.

Roulette simulator.

We can simulate a roulette roll using Excel's =RANDBETWEEN(1,38)function. Here's an example: RouletteWithExcel

But please note: I am not advocating gambling; in fact, I am very much against playing such games as they eventually ruin the gambler. The purpose of this example is to illustrate that roulette is an unfair game and you shouldn't play it with real money!

(d) Topic 4: Confidence Intervals and Hypothesis Testing

d.1 Confidence intervals

Poll Results

Nanos Poll (again!)

Confidence Intervals for the Proportion p

We will do this with the participation of the class and we will use an inflatable globe to estimate the proportion of the water surface to the total surface of the globe.

Here is the video of this experiment I recorded in Section C02 (November 4, 2010, Thursday).

*****

Example: (Population proportion) The CI for population proportion is easy to obtain. Suppose you poll 1000 people and 340 of them state that they would vote Liberal, if the election were held today. Here is what we do to find a 95% CI:

> prop.test(340,1000)

1-sample proportions test with continuity correction

data: 340 out of 1000, null probability 0.5
X-squared = 101.761, df = 1, p-value < 2.2e-16
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
0.3108142 0.3704312
sample estimates:
p
0.34

So, the sample proportion is 0.34, with a 95% CI of [0.3108,0.3704], i.e., a margin of error of about 3%. ¶

d.2 Hypothesis testing

(What is the meaning of the word "hypo" in "hypo-thesis"?

Hypo-allergenic as in ?

Hypo-thermia as in ?

"Hypo-potamus" (??) as in ?                                             Tricked you! This is a hippo-potamus. :-)

"Hypo" means "below, under" in Greek.

"Thesis", is something that is proven to be true.

So, "hypo-thesis" is something that is yet to be proven to be true.

Example: Shishito peppers. It is claimed that 1 in 10 are hot. I am not sure. What if I find 13 hot in a bag of 60? Do the following in R:

> prop.test(13,60,p=0.1)

We'll discuss the results in class.

Example: Lady tasting tea! 

In 2021, Coke and Pepsi with Tina.

Now, what does a lady tasting tea have to do with hypothesis testing?

"The Lady Tasting Tea" : Can tea poured into milk taste differently than that of milk poured into tea? This experiment was originally designed by Professor Ronald Fisher in the 1920s, and it will help us motivate the discussion of hypothesis testing. We will, however, use Coke and Pepsi in our experiment. The "lady" in the story is Dr. Muriel Bristol of Cambridge University.

She claims that she knows the difference. Here, my null hypothesis is "H0: She is guessing". Now, if she is purely guessing, there is a 0.014 probability of getting all 8 cups correct. This is such an unlikely outcome but if it happens, I am willing to change my mind and reject my null H0 and believe that she can tell the difference. But what if she just guessed and got all correct? Then I made a mistake in changing my mind, but the probability of me making this mistake is only 0.014. This is the p-value.

In case you were wondering, here is the mathematics behind the calculations. In this link you can find the probaibilities of 0, 2, 4, 6 or 8 correct identifications which uses the hypergeometric probabilities (which we did not discuss).

Type I and Type II errors

Type I error : In 1959, Steven Truscott was found guilty of murdering his classmate even though he did not commit any crime. In 2007, he was formally acquitted of the crime. In 2008, the government of Ontario awarded him $6.50 million in compensation

Type II error : Many people believe that O. J. Simpson had murdered his wife and he should have been found guilty. But after a lenghty trial, he was acquitted in 1995.

• R documentation for  t.test.
• Background material for hypothesis testing. Wikipedia.

Exercise: For the Atkins problem test the null hypothesis that Atkins users lose, on average, 17 pounds after 6 months. (This is now a two-sided test.)

Exercise: Use the following data values to test the hypothesis that true mean is 750 vs. the hypothesis that it differs from 750: (801,814,784,836,820) What is the p-value? (Answer: p = .0023; so reject the null)