I realize that gambling may be a controversial subject to discuss with the children. Let the record show that I am in no way encouraging them to gamble (quite the opposite is true). Gambling, for better or worse, is a big part of our lives, and I believe it is better to approach it with open eyes and understanding rather than ignorance. Gambling is not limited to what happens in a casino. Playing the lottery and the stock market are also forms of gambling and can provide very different lessons.
I'm trying to take an iterative approach. That is, explain some things, then test, then explain more and test things already tested, plus some new things. I also have ideas for some probabilistic games that they can play. Much of the info here I got from the excellent book "Struck by Lightning: the curious world of probabilities" by Rosenthal
Probability plays a big role in almost everything we do or that happens to us. For example, when you get in a car or an airplane there is a small probability you will be involved in a crash. It is actually good that there is so much randomness, because our lives would be boring without it. Imagine how your life would be if you always knew exactly what was going to happen and how frustrating it would be if there was nothing you could do to change it.
A lot of games involve probability to a greater or lesser degree. Chutes and ladders is purely luck because each move is entirely determined by the roll of a die. Monopoly has a large component of luck, but also involves a surprising amount of skill because of all the decisions that you have to make while playing. Understanding probability will help you be a much better monopoly player. Then there are other games like checkers, chess, and go which have no luck involved. These games are said to have perfect information.
I remember playing monopoly with Brian one time. He had all the railroads and a lot of cash. I had 3 purples, but not much cash. I knew that if play continued with no change, it was a certainty I would lose. So I took a chance and mortgaged all my other properties to buy houses on the purple properties. There was only a small chance of him landing there before I landed on more railroads. But having a small chance of victory quickly (say 25%) if he landed on my houses was better than facing certain defeat in the long run by gradually losing money.
When you shoot a basketball, what is the probability that you make a basket? You can figure it out by trying it 100 times and see how many times you are successful. The number of times you make it divided by the number of times you tried tells you your expected likely hood of success on any given chance.
Let us consider gambling at a casino. It is clearly something that involves a lot of chance. Do you all know about roulette? There is a wheel with 38 spaces for a marble to fall in. Two of the spaces are purple, but the rest are black or red. You can bet on black or red or individual numbers, but if the marble lands in the purple spots the casino wins. So if you bet on red, what is your probability of winning? (18/38 or 47%).
Suppose you borrowed $10,000 from the mob, and if you do not pay it back by tomorrow they are going to break both your legs. You have one night to gamble on roulette in the casino what should you do? Should you make one big bet or lots of little one? why?
Gambling is not something I recommend that you ever do, but if you must you should at least understands the probabilities involved so that at least you will not lose as much as you might otherwise. The casino will always win in the long run even though on any individual wager the can win or loose. The reason they always win is because all the odds are just slightly in their favor and in the long run they will always win a certain percentage of the total amount bet.
If you consider betting 10 dollars on black in roulette, you win about half the time, but because of the 2 purple spaces, on average the hours will take 52 cents out of every 10 dollar wager you make. You could have a fantastic winning streak where you win 10 times in a row, but if you play all night, you are guaranteed to loose approximately 5% of the total amount that you bet.
If Magic Johnson challenged me to a free throw competition and said if I can make more free throws than him in 5 shots at the basket, he will pay me $1000, and if he makes more than him I have to pay him $10. Is this a good bet to make? What about if it were out of 100 shots instead of 5? Should I still take the bet?
If you know that the odds are slightly in your favor, all you have to do is wait and after enough trials you will eventually come out ahead.
Most of the events that we are talking about are independent events. Rolling a dice does not affect what comes up the next time you roll it. If I happen to flip a coin ten times and get heads 10 times, what is the probability that I will get tails on the 11th toss?
[Bonus problem for this week: In a certain country a law was passed that a married couple can have as many girls as they want, but have to stop having children after they have one boy. After 10 years will there be more boys than girls or more girls than boys? Explain.]
What is the probability that you will roll at least one 3 when rolling 4 dice? Solving this involves a common trick. The trick is to find what is the probability of not getting any 6's and then subtracting it from 1. Since you know that you are either going to roll one or more threes or you are not, the total must be one. The probability of not rolling a three is
5/6 * 5/6 * 5/6 * 5/6 = 0.48
That means you are a little more likely to roll at least one three than not roll any. This is actually how the whole branch of mathematics which is probability started. A French gambler in the 17th century was making a nice profit by betting people that at least one 6 would show up if he rolled a dice 4 times. He then changed his bet to saying that a pair of sixes would show in 24 rolls, but because of a miscalculation of the probabilities started losing. He wrote a letter to Blaise Pascal, who wrote a letter to Pierre Fermat (a famous mathematician of the time), and these letters are considered the initial consideration of modern probability theory.
If you calculate the probabilities or just consider the historical results of certain investment activities you can determine an expected return on your investment over the long run. Here are some expected returns.
> Stock market : Some years it goes up some years down. Historically it gains about 10% a year over the long run.
> US treasuries : 2% a year, but does not fluctuate much.
> Roulette : -5% can happen in a single evening.
> Black Jack : -4% if you play very well.
> Slot machines : -10% or -30% percent based on the machines you use.
> State lottery : -40% (-60% if you account for the fact that you will have to pay a lot of tax). You are more likely to die in a car crash driving to the store to get your lottery ticket than you are to win. The people who play the lottery the most are poor people. If Bill Gates invested all his money ($100 billion) in buying lottery tickets every month, in a year he would have 21 million, and after 2 years he would only have 47 thousand left. The government spends millions of dollars a year trying to convince you to buy lottery tickets. Even if you win, about half is kept in taxes.
- Clearly casino gambling is a very poor use of your money if you are trying to hold onto it. So why do people do it? Because it is fun and in some cases addicting.
- why is flipping a coin random? because it is an example of a chaotic system.
- Make 2 sequences of numbers. One truly random, one which is not. Try to guess the difference.
- Play 3 card thriller. Three cards (r/r r/b b/b) bet whether or not the back of the card is red or black. (there is a 2/3 chance its the same as the front, but most will guess that it is a 50-50 chance).
- Play Monte hall puzzle with some sort of real prize. When one of the doors is opened, the remaining door goes from 1/3 to 2/3.
- Things in ordinary life where probability plays a huge role
- insurance
- surveys and polls
- secure internet transactions
- gambling
- genetics
- most computer programs - for example games
I created an applet to simulate the rolling of M dice with N sides (or throwing of M coins).
If you open the options dialog you can select the number of dice and number of sides you want in your simulation.
http://barrybecker4.com/games/dice_en.html
[I may not get to this at all]
Statistics: Branch of math dealing with the analysis and interpretation of numerical data. Much of it is trying to see how much the data is different from random results.
- The death rate in the Navy during the Spanish-American War (1898) was 9 per 1000 For Civilians in New York City during the same period it was 16 per thousand.
Groups are not comparable.
The Navy is made up mostly of young men in known good health
The civilian population includes infants, old, and ill people.
- In one study, 33% of women at Hopkins had married faculty members! There were three women enrolled at the time, and one of them had married a faculty man.
- Explain pi. Do you think that the sequence 0123456789 appears in the number? yes. First around the 500,000 decimal place. But it also appears many more times. An infinite number in fact.
- Chance of getting a heart or a face card (J,Q,K) (or one that is both) is 1/4 + 3/13 - 1/4 * 3/13 = 11/26.
- Causality. Being healthy and doing yoga are highly correlated. Does yoga cause you to be healthy? Probably the set of people that do yoga is not a random sample of the population. It is a self-selecting group. The people who are interesting in trying to have healthy habits take up yoga. Heavy drinkers and smokers are less likely to take up yoga.
Testing for Disease : Suppose you've taken a test for a dreaded disease, which is 99% accurate.
If you have the disease, the test will be positive 99% of the time.
If you don't have the disease, the test will be negative 99% of the time.
Suppose that the disease is rare and that only 0.1% of the population has it.
How concerned should you be if you test positive?
Don't be overly concerned.
You only have 9% chance of having D.
Population of 100,000
0.1% = 100 people have D
99 of the 100 with D will test positive
However, 1% of 99,900 will also test positive, so 999 healthy people
The probability of having D is therefore
99/(999+99) = 9%
For those who understand Bayes Rule : P(D|+) =
P(+|D)*P(D)/(P(+|D)*P(D)+P(+|not D)*P(not D)) =
99%*0.1%/(99%*0.1% + 1%*99.9%) = 9%