Probability Of Slot Machine Calculation

Show from first principles that $P(abland a) = 1$.

Suppose you have a slot machine with three reels with ten symbols on each, and it only pays out when three cherries hit. The odds of winning that jackpot, as we determined earlier, is 1/1000. If we set the jackpot as $900, and charge $1 per bet, the payout percentage for that game will be 90%, or $900/$1000. These results are focused on probability and expected value, these being the most important parameters for decisional criteria in slots. The book is packed with plenty of figures, tables, and formulas. For the most popular categories of slot machines, namely with 3, 5, 9, and 16 reels. Any other category of slot games is covered in the. Founded in 2011, Probability Games can't quite be classed as a brand spanking new name in the industry. Although, having said that, the software company is far from making any huge splashes in the online and mobile slot machine industry, remaining relatively unknown to the vast majority of punters on the web. Compute the probability that playing the slot machine once will result in a win. Estimate the mean and median number of plays you can expect to make until you go broke, if you start with 10 coins. You can run a simulation to estimate this, rather than trying to compute an exact answer. The probability of winning on a slot machine is 5%. If a person plays the machine 500 times, find the probability of winning 30 times. Use the normal approximation to the binomial distribution. Find the area under the normal curve over the interval (29.5,30.5). I've found the solution, thanks to this article on Binomial Probability. It is necessary in this case to think of the slot machine not as 5 reels with 3 positions visible each, but as 15 reels with 1 position visible each. Given: N: The number of symbols we are looking for; P: The probability.

Using the axioms of probability, prove that anyprobability distribution on a discrete random variable must sum to 1.

For each of the following statements, either prove it is true or give acounterexample.

If $P(a b, c) = P(b a, c)$, then$P(a c) = P(b c)$
If $P(a b, c) = P(a)$, then $P(b c) = P(b)$
If $P(a b) = P(a)$, then$P(a b, c) = P(a c)$

Would it be rational for an agent to hold the three beliefs$P(A) {0.4}$, $P(B) {0.3}$, and$P(A lor B) {0.5}$? If so, what range of probabilities wouldbe rational for the agent to hold for $A land B$? Make up a table likethe one in Figure de-finetti-table, and show how itsupports your argument about rationality. Then draw another version ofthe table where $P(A lor B){0.7}$. Explain why it is rational to have this probability,even though the table shows one case that is a loss and three that justbreak even. (Hint: what is Agent 1 committed to about theprobability of each of the four cases, especially the case that is aloss?)

This question deals with the propertiesof possible worlds, defined on page possible-worlds-page as assignments to allrandom variables. We will work with propositions that correspond toexactly one possible world because they pin down the assignments of allthe variables. In probability theory, such propositions are called atomic event. Forexample, with Boolean variables $X_1$, $X_2$, $X_3$, the proposition$x_1land lnot x_2 land lnot x_3$ fixes the assignment of thevariables; in the language of propositional logic, we would say it hasexactly one model.

Prove, for the case of $n$ Boolean variables, that any two distinctatomic events are mutually exclusive; that is, their conjunction isequivalent to ${false}$.
Prove that the disjunction of all possible atomic events islogically equivalent to ${true}$.
Prove that any proposition is logically equivalent to thedisjunction of the atomic events that entail its truth.

ProveEquation (kolmogorov-disjunction-equation) fromEquations (basic-probability-axiom-equation)and (proposition-probability-equation).

Consider the set of all possible five-card poker hands dealt fairly froma standard deck of fifty-two cards.

How many atomic events are there in the joint probabilitydistribution (i.e., how many five-card hands are there)?
What is the probability of each atomic event?
What is the probability of being dealt a royal straight flush? Fourof a kind?

Given the full joint distribution shown inFigure dentist-joint-table, calculate the following:

$textbf{P}({toothache})$.
$textbf{P}({Cavity})$.
$textbf{P}({Toothache}{cavity})$.
$textbf{P}({Cavity}{toothache}lor {catch})$.

Given the full joint distribution shown inFigure dentist-joint-table, calculate the following:

Probability Of Winning A Slot Machine

$textbf{P}({toothache})$.
$textbf{P}({Catch})$.
$textbf{P}({Cavity}{catch})$.
$textbf{P}({Cavity}{toothache}lor {catch})$.

In his letter of August 24, 1654, Pascalwas trying to show how a pot of money should be allocated when agambling game must end prematurely. Imagine a game where each turnconsists of the roll of a die, player E gets a point whenthe die is even, and player O gets a point when the dieis odd. The first player to get 7 points wins the pot. Suppose the gameis interrupted with E leading 4–2. How should the moneybe fairly split in this case? What is the general formula? (Fermat andPascal made several errors before solving the problem, but you should beable to get it right the first time.)

Deciding to put probability theory to good use, we encounter a slotmachine with three independent wheels, each producing one of the foursymbols bar, bell, lemon, orcherry with equal probability. The slot machine has thefollowing payout scheme for a bet of 1 coin (where “?” denotes that wedon’t care what comes up for that wheel):

bar/bar/bar pays 20 coins

bell/bell/bell pays 15 coins

lemon/lemon/lemon pays 5 coins

cherry/cherry/cherry pays 3 coins

cherry/cherry/? pays 2 coins

cherry/?/? pays 1 coin

Compute the expected “payback” percentage of the machine. In otherwords, for each coin played, what is the expected coin return?
Compute the probability that playing the slot machine once willresult in a win.
Estimate the mean and median number of plays you can expect to makeuntil you go broke, if you start with 10 coins. You can run asimulation to estimate this, rather than trying to compute anexact answer. Sky rider slot machine videos.

Deciding to put our knowledge of probability to good use, we encounter aslot machine with three independently turning reels, each producing oneof the four symbols bar, bell,lemon, or cherry with equal probability. Theslot machine has the following payout scheme for a bet of 1 coin (where“?” denotes that we don’t care what comes up for that wheel):

bar/bar/bar pays 21 coins

bell/bell/bell pays 16 coins

lemon/lemon/lemon pays 5 coins

cherry/cherry/cherry pays 3 coins

cherry/cherry/? pays 2 coins

cherry/?/? pays 1 coin

Compute the expected “payback” percentage of the machine. In otherwords, for each coin played, what is the expected coin return?
Compute the probability that playing the slot machine once willresult in a win.
Estimate the mean and median number of plays you can expect to makeuntil you go broke, if you start with 8 coins. You can run asimulation to estimate this, rather than trying to compute anexact answer.

We wish to transmit an $n$-bit message to a receiving agent. The bits inthe message are independently corrupted (flipped) during transmissionwith $epsilon$ probability each. With an extra parity bit sent alongwith the original information, a message can be corrected by thereceiver if at most one bit in the entire message (including the paritybit) has been corrupted. Suppose we want to ensure that the correctmessage is received with probability at least $1-delta$. What is themaximum feasible value of $n$? Calculate this value for the case$epsilon0.001$, $delta0.01$.

Show that the three forms of independence inEquation (independence-equation) are equivalent.

Consider two medical tests, A and B, for a virus. Test A is 95%effective at recognizing the virus when it is present, but has a 10%false positive rate (indicating that the virus is present, when it isnot). Test B is 90% effective at recognizing the virus, but has a 5%false positive rate. The two tests use independent methods ofidentifying the virus. The virus is carried by 1% of all people. Saythat a person is tested for the virus using only one of the tests, andthat test comes back positive for carrying the virus. Which testreturning positive is more indicative of someone really carrying thevirus? Justify your answer mathematically.

Suppose you are given a coin that lands ${heads}$ with probability $x$and ${tails}$ with probability $1 - x$. Are the outcomes of successiveflips of the coin independent of each other given that you know thevalue of $x$? Are the outcomes of successive flips of the coinindependent of each other if you do not know the value of$x$? Justify your answer.

After your yearly checkup, the doctor has bad news and good news. Thebad news is that you tested positive for a serious disease and that thetest is 99% accurate (i.e., the probability of testing positive when youdo have the disease is 0.99, as is the probability of testing negativewhen you don’t have the disease). The good news is that this is a raredisease, striking only 1 in 10,000 people of your age. Why is it goodnews that the disease is rare? What are the chances that you actuallyhave the disease?

After your yearly checkup, the doctor has bad news and good news. Thebad news is that you tested positive for a serious disease and that thetest is 99% accurate (i.e., the probability of testing positive when youdo have the disease is 0.99, as is the probability of testing negativewhen you don’t have the disease). The good news is that this is a raredisease, striking only 1 in 100,000 people of your age. Why is it goodnews that the disease is rare? What are the chances that you actuallyhave the disease?

It is quite often useful to consider theeffect of some specific propositions in the context of some generalbackground evidence that remains fixed, rather than in the completeabsence of information. The following questions ask you to prove moregeneral versions of the product rule and Bayes’ rule, with respect tosome background evidence $textbf{e}$:

Prove the conditionalized version of the general product rule:
Prove the conditionalized version of Bayes’ rule inEquation (conditional-bayes-equation).

Show that the statement of conditional independenceis equivalent to each of the statements

Suppose you are given a bag containing $n$ unbiased coins. You are toldthat $n-1$ of these coins are normal, with heads on one side and tailson the other, whereas one coin is a fake, with heads on both sides.

Suppose you reach into the bag, pick out a coin at random, flip it,and get a head. What is the (conditional) probability that the coinyou chose is the fake coin?
Suppose you continue flipping the coin for a total of $k$ timesafter picking it and see $k$ heads. Now what is the conditionalprobability that you picked the fake coin?
Suppose you wanted to decide whether the chosen coin was fake byflipping it $k$ times. The decision procedure returns ${fake}$ ifall $k$ flips come up heads; otherwise it returns ${normal}$. Whatis the (unconditional) probability that this procedure makes anerror?

Probability Calculations Examples

In this exercise, you will complete thenormalization calculation for the meningitis example. First, make up asuitable value for $P(slnot m)$, and use it to calculateunnormalized values for $P(ms)$ and $P(lnot m s)$(i.e., ignoring the $P(s)$ term in the Bayes’ rule expression,Equation (meningitis-bayes-equation)). Now normalizethese values so that they add to 1.

This exercise investigates the way in which conditional independencerelationships affect the amount of information needed for probabilisticcalculations.

Suppose we wish to calculate $P(he_1,e_2)$ and we have noconditional independence information. Which of the following sets ofnumbers are sufficient for the calculation?
1. ${textbf{P}}(E_1,E_2)$, ${textbf{P}}(H)$,${textbf{P}}(E_1H)$,${textbf{P}}(E_2H)$
2. ${textbf{P}}(E_1,E_2)$, ${textbf{P}}(H)$,${textbf{P}}(E_1,E_2H)$
3. ${textbf{P}}(H)$,${textbf{P}}(E_1H)$,${textbf{P}}(E_2H)$
Suppose we know that${textbf{P}}(E_1H,E_2)={textbf{P}}(E_1H)$for all values of $H$, $E_1$, $E_2$. Now which of the three sets aresufficient?

Let $X$, $Y$, $Z$ be Boolean random variables. Label the eight entriesin the joint distribution ${textbf{P}}(X,Y,Z)$ as $a$ through$h$. Express the statement that $X$ and $Y$ are conditionallyindependent given $Z$, as a set of equations relating $a$ through $h$.How many nonredundant equations are there?

(Adapted from Pearl [-@Pearl:1988].) Suppose you are a witness to anighttime hit-and-run accident involving a taxi in Athens. All taxis inAthens are blue or green. You swear, under oath, that the taxi was blue.Extensive testing shows that, under the dim lighting conditions,discrimination between blue and green is 75% reliable.

Is it possible to calculate the most likely color for the taxi?(Hint: distinguish carefully between the propositionthat the taxi is blue and the proposition that itappears blue.)
What if you know that 9 out of 10 Athenian taxis are green?

Write out a general algorithm for answering queries of the form${textbf{P}}({Cause}textbf{e})$, using a naive Bayesdistribution. Assume that the evidence $textbf{e}$ may assign values toany subset of the effect variables.

Text categorization is the task ofassigning a given document to one of a fixed set of categories on thebasis of the text it contains. Naive Bayes models are often used forthis task. In these models, the query variable is the document category,and the “effect” variables are the presence or absence of each word inthe language; the assumption is that words occur independently indocuments, with frequencies determined by the document category.

Explain precisely how such a model can be constructed, given as“training data” a set of documents that have been assignedto categories.
Explain precisely how to categorize a new document.
Is the conditional independence assumption reasonable? Discuss.

In our analysis of the wumpus world, we used the fact thateach square contains a pit with probability 0.2, independently of thecontents of the other squares. Suppose instead that exactly $N/5$ pitsare scattered at random among the $N$ squares other than [1,1]. Arethe variables $P_{i,j}$ and $P_{k,l}$ still independent? What is thejoint distribution ${textbf{P}}(P_{1,1},ldots,P_{4,4})$ now?Redo the calculation for the probabilities of pits in [1,3] and[2,2].

Redo the probability calculation for pits in [1,3] and [2,2],assuming that each square contains a pit with probability 0.01,independent of the other squares. What can you say about the relativeperformance of a logical versus a probabilistic agent in this case?

Implement a hybrid probabilistic agent for the wumpus world, based onthe hybrid agent inFigure hybrid-wumpus-agent-algorithm and theprobabilistic inference procedure outlined in this chapter.