## Monty Hall problem and Two envelopes problem

### August 29, 2009

Monty Hall problem.
This problem became popular after the famous american show, where a participant at random choose one of three doors. Behind one door there is a car, behind others there are goats. Host knows where a car is and after your choosing leave your door closed, then opens the door with a goat from one of two left doors. And now the question from the host to a player: “Would you change the door?”.

For example, you chose first door. The host opened second door and show you a goat, then asked you to stay with the first door or change to the third.

Probability says yes, you should change! There are many answers to this problem in the internet, but they are so complicated, I try to make it simple. Look at the probability tree:

If you choose strategy “Stay” you win 1 car and 2 goats.
If you choose strategy “Change” you win 2 cars and 1 goat.
So it is better to change, you raise your chances to win!

Two envelopes problem.
There are two envelopes with money. In one envelope the sum is as twice as the sum in another. You do not know where is the envelope with the twice sum. Then you choose at random an envelope and open it, see the sum. Would you take the second envelope instead of the opened? For example, you see $10, now if you change you can get$5 or $20 in the second envelope or stay with$10.

There appears the problem.

1. Consider the case when there are two sum of money $10 and$20 in the envelopes, look at the picture what is happening when you make decision:

As you can see there is no deference.

2. Consider another case were you see $10 in the envelope, hence in the second envelope there may be$5 or $20: Expected value is $P(stay)=1/2 * 10=5, ~~ P(change)=1/4 * 5+1/4 * 20 = 7.5.$ So in this example it is better to change and your gain will be 1.5 times more with change strategy. The secret is that if you change you can lose$5 or add $10, what is not the same. Simple program on C# shows next results for two considered methods (at every play there are random money in the envelopes): In this simulation ultimate sums are approximately equal in the first method, in the second method if you change envelope you will raise the sum. Advertisements ## Slot machines ### August 25, 2009 Brief history of a slot machine. In the late 19th century first slot machines were developed. It had a wheel with different pictures and a handle for start playing. Player gave money to a bartender and tried to get winning picture on a rotated wheel. In 1985 Charles Fey, a car mechanic, introduced new slot machine – The Liberty Bell slot machine. It had 3 reels and 10 pictures on each reel. The name was given due to winning combination, the highest pay-off was with 3 bell pictures on the reels stopped in a row. In 1907 Herbert Mills started mass production of new slot machines, where player could put a coin in a machine and play. Mills added famous fruit symbols. It was so convenient to play and there was not need any high knowledge that game had wide spread. Then machines were advanced, there became 20, then 22 signs on a reel. In mid 80th last century appeared slot machines with merely computer chips without evaluating a win by rotated mechanism. And they are working now. How does it work? There is a chip with random number generator. The cheap generates 3 numbers and then gets a picture from a programmed table. Consider the huge table in Red White & Blue slot machine: If the machine’s chip generates three numbers 57, 59, 57. It means that computer orders to a motor to rotate reels and on the first reel will be “2 bar”, on the second reel will be “1 bar”, on the third reel will be “blank”. Also there can be LCD panels without real reels. What are the pay-offs? On the next table there are information how often each picture appears: Now we can calculate probabilities to win and to lose, for instance. the probability that there can be three white 7 in a row is $P(E)=\frac{6*1*7}{64*64*64}=0.000011:$ The probability to lose is $P(lose)=0.826.$ It means that at 1000 attempts to win at average in long run: a player loses 826 times and wins 174 times. However, in long run you can back your money with the probability $P(money)=0.865.$ That is what the producer of the machine has published. If each attempt does not depend on any previous attempt, then one could claim that at every time odds to win are equal. No, they are not. Consider example with a coin. If you bet on Tails, what is the probability that there will be 6 Heads? The probability that there will be 6 Heads is $\frac{1}{64}$ and the probability that in all 6 tosses there will be at least 1 Tails is $\frac{63}{64}.$ At the same time the probability that there will be Tails or Heads in each attempt is $\frac{1}{2},$ because of independent events. So it looks as moot point. Think, it is clear in statistic analyse that if you go to play 40 times in Red White & Blue machine the probability that you win at least one time is $P(10)=1- (0.8265)^{10}=0.851.$ However if somebody played and had 50 loses and you sat to play after him, and you played 10 times too. The probability that you win at least one time in 60 attempts is $P(60)=1- (0.8265)^{60}=0.999.$ If the machine is not cheating you can raise your odds to win, though there are random win in every attempt. It looks silly if manufactures do not manipulate the slot machines and produce them fair. The best way to get good money from any game is to make addiction. Good way to do it is to give to a player initially frequent small pay-off and then rare large pay-off. However with fair random chip a player is able to win initially large pay-off and so will not be interested in subsequent games. One more point where slot machines owners can manipulate is the highest pay-off. For a usual player amount$1 million to win is huge, $3 million is huge and$20 million is huge, usual player does not feel the real difference. Thus it is better to give 3 times to win $3 million and promote wide advertisement with winners than give 1 time to win$40 million.

In mid 50s 50% revenues from all casinos in state Nevada were made by slot machines. In late 80s this were up to 70%. Today a slot machine is one of the best way to make profit .

Statistic data: The Wizards of Odds, Slot Machines (http://wizardofodds.com/slots).