AP Statistics Curriculum 2007 Prob Simul - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T20:37:45Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Dinov: /* Chuck A Luck Experiment */

2014-09-15T13:40:39Z

‎Chuck A Luck Experiment

Dinov: /* Chuck A Luck Experiment */

2014-09-15T13:40:17Z

‎Chuck A Luck Experiment

IvoDinov: /* Chuck A Luck Experiment */

2013-04-17T01:50:00Z

‎Chuck A Luck Experiment

IvoDinov: /* Chuck A Luck Experiment */

2013-04-17T01:48:04Z

‎Chuck A Luck Experiment

IvoDinov at 00:16, 2 November 2012

2012-11-02T00:16:43Z

IvoDinov: moved the Poker Game Counting and probablities section to Counting section

2011-04-25T15:46:54Z

moved the Poker Game Counting and probablities section to Counting section

Jenny at 16:55, 28 June 2010

2010-06-28T16:55:44Z

IvoDinov: /* Poker Game */

2009-11-19T17:02:15Z

‎Poker Game

IvoDinov: added a link to the Problems set

2009-10-26T04:50:29Z

added a link to the Problems set

← Older revision		Revision as of 20:37, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.~~stat~~.~~ucla~~.edu~~/socr~~/index.php?title=AP_Statistics_Curriculum_2007_Prob_Simul}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Prob_Simul}}

@@ Line 56: / Line 56: @@
 Similarly, we can compute the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Variance|variance]] and [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Standard_Deviation|standard deviation]] of the outcome/payoff variable X:
-:  \( Var[W]= \( E[W]=\sum_w{(w-E[W])^2P(W=w)} = 1.23917\) and \( SD[W]= \sqrt{Var[W]} = 1.11318\).
+:  \( Var[W]= \sum_w{(w-E[W])^2P(W=w)} = 1.23917\) and \( SD[W]= \sqrt{Var[W]} = 1.11318\).
 : '''Question''': What is your interpretation of this quantitative analysis of the [http://socr.ucla.edu/htmls/exp/Chuck_A_Luck_Experiment.html Chuck A Luck Experiment]? Would you agree to play (or host) this game?

@@ Line 56: / Line 56: @@
 Similarly, we can compute the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Variance|variance]] and [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Standard_Deviation|standard deviation]] of the outcome/payoff variable X:
-:  \( Var[W]= 1.23917\) and \( SD[W]= 1.11318 = \sqrt{Var[W]}\).
+:  \( Var[W]= \( E[W]=\sum_w{(w-E[W])^2P(W=w)} = 1.23917\) and \( SD[W]= \sqrt{Var[W]} = 1.11318\).
 : '''Question''': What is your interpretation of this quantitative analysis of the [http://socr.ucla.edu/htmls/exp/Chuck_A_Luck_Experiment.html Chuck A Luck Experiment]? Would you agree to play (or host) this game?

@@ Line 44: / Line 44: @@
 | Outcome = Number of matching dice  || None=loss || 1 || 2 || 3
 |-
-| X (profit/payoff)  || -1 || 1 || 2 || 3
+| W (profit/payoff)  || -1 || 1 || 2 || 3
 |-
-| P(X=x) || \( \left( \frac{5}{6} \right)^3 \) || \(\binom{3}{1} \left( \frac{5}{6} \right)^2 \frac{1}{6}\) || \(\binom{3}{2}\frac{5}{6} \left( \frac{1}{6} \right)^2 \) || \( \left(\frac{1}{6}\right)^3 \)
+| P(W=w) || \( \left( \frac{5}{6} \right)^3 \) || \(\binom{3}{1} \left( \frac{5}{6} \right)^2 \frac{1}{6}\) || \(\binom{3}{2}\frac{5}{6} \left( \frac{1}{6} \right)^2 \) || \( \left(\frac{1}{6}\right)^3 \)
 |-
-| x*P(X=x) || (-1)x0.5787 || 0.34722 || 2x0.06944  || 3x0.00463
+| w*P(W=w) || (-1)x0.5787 || 0.34722 || 2x0.06944  || 3x0.00463
 |}
 </center>
-Thus, if X is the [[AP_Statistics_Curriculum_2007_Distrib_RV |random variable]] denoting the return of this game (player's profit), its expected value is:
+Thus, if W is the [[AP_Statistics_Curriculum_2007_Distrib_RV |random variable]] denoting the return of this game (player's profit), its expected value is:
-: \( E[X]=\sum_x{xP(X=x)} = -0.0787\).
+: \( E[W]=\sum_w{wP(W=w)} = -0.0787\).
 Similarly, we can compute the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Variance|variance]] and [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Standard_Deviation|standard deviation]] of the outcome/payoff variable X:
-:  \( Var[X]= 1.23917\) and \( SD[X]= 1.11318 = \sqrt{Var[X]}\).
+:  \( Var[W]= 1.23917\) and \( SD[W]= 1.11318 = \sqrt{Var[W]}\).
 : '''Question''': What is your interpretation of this quantitative analysis of the [http://socr.ucla.edu/htmls/exp/Chuck_A_Luck_Experiment.html Chuck A Luck Experiment]? Would you agree to play (or host) this game?

@@ Line 33: / Line 33: @@
 ====[[SOCR_EduMaterials_Activities_ChuckALuckExperiment#Probability_calculations| Chuck A Luck Experiment]]====
-The [http://socr.ucla.edu/htmls/exp/Chuck_A_Luck_Experiment.html Chuck A Luck Experiment] shows how to compute the expectation of a real game, which indicated the odds of a player to win or loose money.
+The [http://socr.ucla.edu/htmls/exp/Chuck_A_Luck_Experiment.html Chuck A Luck Experiment] shows how to compute the expectation of a real game, which indicated the odds of a player to win or loose money. In this game, a player/gambler chooses an integer from ''1'' to ''6''. Then three fair dice are rolled. If exactly ''k'' dice show the gambler's number, the payoff is ''k:1''. The random variable W represents the gambler's net profit on a ''$1'' bet.
 Let's first look at the outcomes of each experiment and compute the corresponding outcome probability values. Then, we can use the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Expectation_.28Mean.29 |expectation formula]] to compute the expected return (expected value) for this experiment. A zero, positive or negative expected return indicates if the game is fair, or biased for the player or the house, respectively.

← Older revision		Revision as of 00:16, 2 November 2012
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	<center>[[Image:SOCR_EBook_Dinov_Probability_012908_Fig7.jpg\|400px]]</center>		<center>[[Image:SOCR_EBook_Dinov_Probability_012908_Fig7.jpg\|400px]]</center>
		+
		+	====[[SOCR_EduMaterials_Activities_ChuckALuckExperiment#Probability_calculations\| Chuck A Luck Experiment]]====
		+	The [http://socr.ucla.edu/htmls/exp/Chuck_A_Luck_Experiment.html Chuck A Luck Experiment] shows how to compute the expectation of a real game, which indicated the odds of a player to win or loose money.
		+
		+	Let's first look at the outcomes of each experiment and compute the corresponding outcome probability values. Then, we can use the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Expectation_.28Mean.29 \|expectation formula]] to compute the expected return (expected value) for this experiment. A zero, positive or negative expected return indicates if the game is fair, or biased for the player or the house, respectively.
		+
		+	The table below identifies the possible outcomes of this experiment and shows how to use the [[AP_Statistics_Curriculum_2007_Distrib_Binomial#Binomial_Random_Variables\|Binomial Distribution]] to compute their corresponding probabilities. Note that the probability of a single die turning up and matching (or not matching, the [[AP_Statistics_Curriculum_2007_Prob_Basics\|complementary event]]) the player betting number is \(\frac{1}{6}\) (or \(\frac{5}{6}\)).
		+
		+	<center>
		+	{\| class="wikitable" style="text-align:center; width:75%" border="1"
		+	\|-
		+	\| Outcome = Number of matching dice \|\| None=loss \|\| 1 \|\| 2 \|\| 3
		+	\|-
		+	\| X (profit/payoff) \|\| -1 \|\| 1 \|\| 2 \|\| 3
		+	\|-
		+	\| P(X=x) \|\| \( \left( \frac{5}{6} \right)^3 \) \|\| \(\binom{3}{1} \left( \frac{5}{6} \right)^2 \frac{1}{6}\) \|\| \(\binom{3}{2}\frac{5}{6} \left( \frac{1}{6} \right)^2 \) \|\| \( \left(\frac{1}{6}\right)^3 \)
		+	\|-
		+	\| x*P(X=x) \|\| (-1)x0.5787 \|\| 0.34722 \|\| 2x0.06944 \|\| 3x0.00463
		+	\|}
		+	</center>
		+
		+	Thus, if X is the [[AP_Statistics_Curriculum_2007_Distrib_RV \|random variable]] denoting the return of this game (player's profit), its expected value is:
		+	: \( E[X]=\sum_x{xP(X=x)} = -0.0787\).
		+
		+	Similarly, we can compute the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Variance\|variance]] and [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Standard_Deviation\|standard deviation]] of the outcome/payoff variable X:
		+	: \( Var[X]= 1.23917\) and \( SD[X]= 1.11318 = \sqrt{Var[X]}\).
		+
		+	: '''Question''': What is your interpretation of this quantitative analysis of the [http://socr.ucla.edu/htmls/exp/Chuck_A_Luck_Experiment.html Chuck A Luck Experiment]? Would you agree to play (or host) this game?

	===[[EBook_Problems_Prob_Simul\|Problems]]===		===[[EBook_Problems_Prob_Simul\|Problems]]===

← Older revision		Revision as of 15:46, 25 April 2011
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	<center>[[Image:SOCR_EBook_Dinov_Probability_012908_Fig4.jpg\|400px]]</center>		<center>[[Image:SOCR_EBook_Dinov_Probability_012908_Fig4.jpg\|400px]]</center>
−
−
−	~~====[[SOCR_EduMaterials_Activities_PokerExperiment \| Poker Game]]====~~
−	There is a variety of events that may be of interest for 5-card (poker) games (see also [[AP_Statistics_Curriculum_2007_Prob_Count#Hands-on_combination_activity \| this section on card games]]. Some of these are:
−	* '''A single Pair Match''': For instance, the hand with the pattern AABCD, where A, B, C and D are from the distinct kinds (denominations) of cards: aces, twos, threes, tens, jacks, queens, and kings (there are 13 denominations, and four suits, in the standard 52 card deck). The number of such hands is <math>{13 \choose 1}{4\choose 2}{12\choose 3}{4\choose 1}^3</math>. If all hands are equally likely, the probability of a single pair is obtained by dividing this number by the total number of 5-card hands possible (<math>{52\choose 5}=2,598,960</math>). Thus, P(1 pair only) = 0.422569.
−
−	* '''Two pairs''': For instance, the pattern AABBC where A, B, and C are from distinct kinds. The number of such hands is <math>{13\choose 2}{4\choose 2}{4\choose 2}{11\choose 1}{4\choose 1}</math>. And therefore, dividing by <math>{52\choose 5}=2,598,960</math>, the P(2 pairs) = 0.047539.
−
−	* '''One triple''': For example, the pattern AAABC where A, B, and C are from distinct kinds. The number of such hands is <math>{13\choose 1}{4\choose 3}{12\choose 2}{4\choose 1}^2</math>. Thus, the P(1 triple) = 0.021128.
−
−	* '''A Full House''': This includes patterns like AAABB where A and B are from distinct kinds. The number of such hands is <math>{13\choose 1}{4\choose 3}{12\choose 1}{4\choose 2}</math>. Thus, P(Full House)=0.001441.
−
−	* '''Four of a kind''': For instance, the pattern AAAAB where A and B are from distinct kinds. The number of such hands is <math>{13\choose 1}{4\choose 4}{12\choose 1}{4\choose 1}</math>. The P(4-of-a-kind)=0.000240.
−
−	* '''A straight''': Five cards in a sequence (e.g., 4,5,6,7,8), with aces allowed to be either 1 or 13 (low or high) and with the cards allowed to be of the same suit (e.g., all hearts) or from some different suits. The number of such hands is <math>10{4\choose 1}^5</math>. Thus, P(A Straight)=0.003940. But if you exclude Straight-Flushes AND Royal Flushes, the number of such hands is <math>10{4\choose 1}^5 - 36 - 4 = 10200</math>, the corresponding probability P(A Straight, but not a Straight or Royal Flush)=0.00392465.
−
−	* '''A Straight Flush''': All 5 cards are from the same suit and they form a straight. The number of such hands is 4*10, and P(Straight Flush)=0.0000153908.
−
−	* '''A Royal Flush''': This consists of the ten, jack, queen, king, and ace all in one suit. There are only 4 such hands. Thus, P(Royal Flush)=0.00000153908.
−
−	~~<center>[[Image:SOCR_EBook_Dinov_Probability_013008_Fig2.jpg\|400px]]</center>~~

	====[[SOCR_EduMaterials_Activities_RouletteExperiment \| Roulette Experiment]]====		====[[SOCR_EduMaterials_Activities_RouletteExperiment \| Roulette Experiment]]====

@@ Line 10: / Line 10: @@
 ===SOCR simulations===
-There are a large number of [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Simulations] that may be used to compute (approximately) probabilities of various processes and compare these empirical probabilities to their exact counterparts.
+There is a large number of [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Simulations] that may be used to compute (approximately) probabilities of various processes and compare these empirical probabilities to their exact counterparts.
 ====[[SOCR_EduMaterials_Activities_BallAndRunExperiment | Ball and Urn Experiment]]====
-This experiment allows sampling with or without replacement from a virtual urn that has one of two types of balls - red (successes) and green (failures). The user may control the total number of balls in the urn (N), the number of red balls (R) and the number of balls sampled from the urn (n). Depending on whether we sample with or without replacement, the chance (or probability) of getting m red balls (successes) in a sample of n balls changes. Note that when sampling ''without replacement'', <math>1 \leq n \leq N</math>, and when sampling ''with replacement'', <math>1 \leq n</math>. The applet records numerically the empirical outcomes and compares these to the theoretical expected counts using distribution tables and graphs. Suppose we set N=50, R=25 and n=10. How many successes (red ball) do we expect to get in the sample of 10 balls, if we sample without replacement? [[SOCR_EduMaterials_Activities_Binomial_Distributions | Hypergeometric distribution]] provides the theoretical model for experiment and allows us to compute this probability exactly. We can also run the experiment once and approximate this answer by dividing the number of observed red balls by 10 (the sample size). You can try this experiment gauge the accuracy of the simulation-based approximations relative to various setting, e.g., sample size. According to the [[SOCR_EduMaterials_Activities_LawOfLargeNumbers | Law of Large Numbers]], the accuracy of the estimation will rapidly increase with the increase of the sample-size (n).
+This experiment allows sampling with or without replacement from a virtual urn that has one of two types of balls - red (successes) and green (failures). The user may control the total number of balls in the urn (N), the number of red balls (R) and the number of balls sampled from the urn (n). Depending on whether we sample with or without replacement, the chance (or probability) of getting m red balls (successes) in a sample of n balls changes. Note that when sampling ''without replacement'', <math>1 \leq n \leq N</math>, and when sampling ''with replacement'', <math>1 \leq n</math>. The applet records numerically the empirical outcomes and compares these to the theoretical expected counts using distribution tables and graphs. Suppose we set N=50, R=25 and n=10. How many successes (red ball) do we expect to get in the sample of 10 balls, if we sample without replacement? [[SOCR_EduMaterials_Activities_Binomial_Distributions | Hypergeometric distribution]] provides the theoretical model for experiment and allows us to compute this probability exactly. We can also run the experiment once and approximate this answer by dividing the number of observed red balls by 10 (the sample size). You can try this experiment to gauge the accuracy of the simulation-based approximations relative to various setting, e.g., sample size. According to the [[SOCR_EduMaterials_Activities_LawOfLargeNumbers | Law of Large Numbers]], the accuracy of the estimation will rapidly increase with the increase of the sample-size (n).
 <center>[[Image:SOCR_EBook_Dinov_Probability_012908_Fig6.jpg|400px]]</center>

@@ Line 29: / Line 29: @@
 ====[[SOCR_EduMaterials_Activities_PokerExperiment | Poker Game]]====
-There is a variety of events that may be of interest for 5-card (poker) games. Some of these are:
+There is a variety of events that may be of interest for 5-card (poker) games (see also [[AP_Statistics_Curriculum_2007_Prob_Count#Hands-on_combination_activity | this section on card games]]. Some of these are:
 * '''A single Pair Match''': For instance, the hand with the pattern AABCD, where A, B, C and D are from the distinct kinds (denominations) of cards: aces, twos, threes, tens, jacks, queens, and kings (there are 13 denominations, and four suits, in the standard 52 card deck). The number of such hands is <math>{13 \choose 1}{4\choose 2}{12\choose 3}{4\choose 1}^3</math>. If all hands are equally likely, the probability of a single pair is obtained by dividing this number by the total number of 5-card hands possible (<math>{52\choose 5}=2,598,960</math>). Thus, P(1 pair only) = 0.422569.

← Older revision		Revision as of 04:50, 26 October 2009
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	<center>[[Image:SOCR_EBook_Dinov_Probability_012908_Fig7.jpg\|400px]]</center>		<center>[[Image:SOCR_EBook_Dinov_Probability_012908_Fig7.jpg\|400px]]</center>
		+
		+	===[[EBook_Problems_Prob_Simul\|Problems]]===

	<hr>		<hr>