1. Introduction …………………………………………………………………21.1. Probability ………………………………………………………...2

1.1.1. Experimental .................................................................21.1.2. Theoretical .....................................................................31.1.3. Conditional .....................................................................4

1.2. Sampling with and without replacement.....................62. Rationale ..................................................................................................73. Modeling ..................................................................................................8

3.1. Game ...........................................................................................83.1.1. Objective...........................................................................83.1.2. Rules...................................................................................8

3.2. Outcome ....................................................................................83.2.1. Raw data ..........................................................................83.2.2. Proccessed data .........................................................13

4. Conclusion ............................................................................................175. References ............................................................................................18

TABLE OF CONTENTS

1.INTRODUCTION

1.1 ProbabilityProbability is the extent to which something is probable; the likelihood of something

happening or being the case.1 By using mathematics, one can describe the chance of an event happening.2 Probability in math is a number between 0 and 1 which describes the odds of a certain event occurring. An impossible event has 0% probability of happening and a certain event has 100% probability of happening3.

One can calculate probability either by observing results of an experiment (experimental probability) or by using “arguments of symmetry” (theoretical probability)4.

1.1.1 Experimental probability

In experiments, there are 4 key terms that are used to calculate probability:

Number of trials: number of times the test has been conducted Outcomes: the different results for each trial of the test Frequency: of a specific outcome is the number of times that outcome has been

observed Relative frequency: of an outcome is the frequency expressed as a fraction of

percentage for total number of trials

To make the terms clearer, an example experiment has ben conducted; a coin has been thrown 200 times. The outcome can be either heads or tails. In the table below is the recorded data.

Table 1.

The relative frequency or probability is calculated by

1„Probability - Definition and More from the Free Merriam-Webster Dictionary“, Merriam-Webster, 2014, <http://www.merriam-webster.com/dictionary/probability> (28.9.2014.)2 Haese, Robert, Sandra Haese, Michael Haese, Marjut Maenpaa, and Mark Humphries. Mathematics for the International Student: Mathematics SL. Adelaide: haese Mathematics, 2012. Print.3 Ibid.4 Ibid.

2 | P a g e

OUTCOMES

Probability (outcome A )= frequency of outcome Anumber of trials

P (heads )=109200

=0.545×100%=54.5%

P (heads )= 91200

=0.455×100%=45.5%

This ensues that the probability for flipping a coin and it landing on heads is 54.5% and for it landing on tails 45.5%. This is the relative frequency. From no further testing, the only conclusion one can pull is that these are the odds of a coin flip. But, if one were to have a bigger number of trials, they would observe chances closer to 50% for both heads and tails5.

1.1.2 Theoretical probability

This probability is based on what we theoretically expect to occur.6 The chance for any outcome to happen is the equal. An example, the die. A die has 6 sides ergo 6 outcomes

and if we assume that the die is not loaded, every side (number) has a 16 chance. And so the

same formula can be used:

Probability (outcome A )= frequency of outcome Anumber of trials

In theoretical probability there are complementary and compound events.

Complementary events are those were one of the events must occur;

P (A )+P (A ´ )=1

Considering if A is an event, A´ is the respective complementary event.

In the example of a coin;

P (heads )+P (heads´ )=1

12+ 12=1

5 „Lawoflargenumbersanimation2.gif (100×169) “, Wikimedia, unknown, <http://upload.wikimedia.org/wikipedia/commons/4/49/Lawoflargenumbersanimation2.gif> (28.9.2014.)6 Haese, Robert, Sandra Haese, Michael Haese, Marjut Maenpaa, and Mark Humphries. Mathematics for the International Student: Mathematics SL. Adelaide: haese Mathematics, 2012. Print.

3 | P a g e

Compound events are the probability of two or more things happening at once7. These kinds of experiments a conducted with two or more object for example a coin and a die, or two coins or two dies. There are independent compound events and dependent compound events.

Independent compound events are those where one event does not affect the probability of the second, third, nth event.8 As an example, tossing a die does not effect a coin in any way so the two events are independent of each other and each have their own probability. To calculate the independent events, saying that A is one event and B is another, one uses this formula;

P (A∧B )=P (A )×P (B)

For example, to calculate what is the probability of getting heads on a coin and rolling an even number on a die;

P (heads∧anevennumber )=P (heads )× P ( evennumber )=12× 36= 312

= 14=25%

Dependent compound events are those were the outcome of one event affect the second, third, nth event9. An example for this is playing cards. There are 52 cards, minus jokers, in one deck of cards. If you pull out one card, and then another, without replacing or putting the first card back, what are the chances to pull out an ace and a 5? The general formula is;

P (A∧B )=P (A )×P (B∨A)

P (anace∧a5 )=P (ace)×P (a5|ace )= 452× 451

= 4663

=0.60%

After pulling out one card, that reduces the number or cards in the deck so the second card has a slightly bigger chance to be any other card.

1.1.3 Conditional probability

The conditional probability of an event is the possibility that one event will occur after another event that has already occurred.10 It is denoted as A∨B read as “A given that B”.

7 „www.shmoop.com/basic-statistics-probability/compound-events.html“, Basic Statistics and Probability, 2014 <http://www.shmoop.com/basic-statistics-probability/compound-events.html> (28.9.2014)8 Haese, Robert, Sandra Haese, Michael Haese, Marjut Maenpaa, and Mark Humphries. Mathematics for the International Student: Mathematics SL. Adelaide: haese Mathematics, 2012. Print.9 „Dependent Events“, Math Goodies, 2014, <http://www.mathgoodies.com/lessons/vol6/dependent_events.html> (28.9.2014)10„Conditional probability - Wikipedia, the free encyclopedia“, Wikipedia, <http://en.wikipedia.org/wiki/Conditional_probability> (1.11.2014)

4 | P a g e

A∨B=P ( A∩B )P (B )

An example; there are 20 jars of marmalade, 9 out of strawberry, 7 out of blueberry and 4 out of raspberry. 13 jars are on the first shelf (6 strawberry, 4 blueberry, 3 raspberry) and 7 are on the second one (3 strawberry, 3 blueberry, 1 raspberry). What are the chances to end up with a blueberry jar? A tree diagram will help with determining this.

First, we must calculate what is the chance to pick a blueberry jar.

P (blueberry jar )=[ 1320× 413+ 720 × 37 ]×100%=[0.199+0.149 ]×100%=34.8%

Next, we have to create a condition. The condition is that the blueberry jar has to be on the first shelf.

P (blueberry jar on first shelf )=[ 1320× 413 ]×100%=19.9%

Next, we calculate what are the chances to pick a blueberry jam jar after we have chosen to look on the first shelf.

P (first shelf|blueberry jam )=P (blueberry jar on first shelf )P (b lueberry jar )

=0.1990.348

=0.571=57.1%

So, the conditional probability of picking a blueberry jam jar after we have chosen to look on the first shelf is 57.1%

1.2 Sampling with and without replacement

5 | P a g e

FIRSTSHELF

SECONDSHELF

1320

720

SJ

SJ

BJ

BJ

RJ

RJ

613

413

313

37 3

717

Sampling is choosing one object at random out of a group of many. This technique is used mostly in checking quality at major factories to ensure that standards are high. 11 One can sample with and without replacement. An example:

From a hat containing 4 names, Linda 5 times, Tom 4 times, Jackie 6 times and Rodger 2 times, you must pick two to do two chores around the house. If you were to sample with replacement, you would pull a name out of the hat, that person would get a chore, and then put his name back in, making his chances bigger to be picked again and the others less. Without replacement, you would not put his name back in the hat, but would choose another one minus one name. In the case with replacement, the events are independent; no matter what name you choose, everyone has the same chances the first and the second time. But, with replacements, you slightly change the chances for everyone involved the second time by minimizing the pool of choices.

2.RATIONALE

11 Haese, Robert, Sandra Haese, Michael Haese, Marjut Maenpaa, and Mark Humphries. Mathematics for the International Student: Mathematics SL. Adelaide: haese Mathematics, 2012. Print.

6 | P a g e

The reason why the topic of probablitity and sampling has been taken for this investigation is because the topic of calculating chances is very interesting to me. Blackjack counters use their witt and probability knowledge to remeber the cards and figure out how to play. And in everyday bets and card games, the speed of calucating the probability of a certain card getting tagged out is the key to winning or losing. Overall, I am interested how to calculate the chances of certain events, faster and possibly in my mind.

The modeling I chose was to play a game that included sampling without replacment. The playing material is 8 buttons, 5 green ones and and 3 red ones. A red button is worth 3 points and the green one 1 point. The players will 10 rounds for 1 game. To win in a round, you have to get 10 points before the other player does. The overall winner is the one with the most rounds won.

The aim of the invesitgation is for me to create a better understanding of real life statistics and its use.

3.MODELING

3.1 Game7 | P a g e

3.1.1. Objective

Two player game with 10 rounds, more rounds won, overall winner. Each player gets a turn to draw 5 buttons from a hat. There are 3 red buttons worth 3 points and 5 green buttons worth 1 point. After every round, the player that has gained 10 points in a smaller amount of draws wins. There are 10 rounds played, the player with more rounds won is the overall winner.

3.1.2. Rules

Per a round, a player can draw five times. First one player draws all of their buttons, puts the buttons back, followed by the second player. The winner of the round is the one who gets the 10 points faster i.e. in less draws. The player with more rounds won after 10 rounds, wins overall. If a player does not get 10 points in a round after the five draws, he loses the round. If both players lose the round, it is a tie. If both players draw 10 points, the one with less times drawn wins.

3.2 Outcome

3.2.1. Raw data

I played against my friend Bella and these are the following results.

ROUND 1

SARA1 12 13 34 15 3

ROUND? none

BELLA1 12 13 34 35 1

ROUND? none

WINNER TIE

ROUND 2SARA

1 12 3

8 | P a g e

3 14 15 3

ROUND? none

BELLA1 12 33 14 35 1

ROUND? none

WINNER TIE

ROUND 3SARA

1 12 33 14 35 3

ROUND? 5

BELLA1 32 13 34 15 1

ROUND? none

WINNER SARA

ROUND 4SARA

1 12 13 34 35 1

ROUND? none

BELLA1 32 13 1

9 | P a g e

4 15 3

ROUND? none

WINNER TIE

ROUND 5SARA

1 32 13 14 35 1

ROUND? none

BELLA1 32 13 14 35 1

ROUND? none

WINNER TIE

ROUND 6SARA

1 32 13 14 35 1

ROUND? none

BELLA1 32 33 14 35 1

ROUND? 4

WINNERBELL

A

ROUND 7SARA

1 1

10 | P a g e

2 13 34 15 3

ROUND? none

BELLA1 32 13 14 15 3

ROUND? none

WINNER TIE

ROUND 8SARA

1 32 13 34 15 3

ROUND? 5

BELLA1 32 13 34 15 3

ROUND? 5

WINNER TIE

ROUND 9SARA

1 12 33 14 15 3

ROUND? none

BELLA1 12 1

11 | P a g e

3 34 35 3

ROUND? 5

WINNERBELL

A

ROUND 10SARA

1 12 33 14 35 3

ROUND? 5

BELLA1 12 33 34 35 1

ROUND? 4

WINNERBELL

A

3.2.1. Proccessed data

12 | P a g e

What are the chances that Bella won in the three rounds that she won?

13 | P a g e

CHOOSE A BUTTON

RED BUTTON

RED BUTTON

RED BUTTON

GREEN BUTTON

GREEN BUTTON

GREEN BUTTON

RED BUTTON

GREEN BUTTON

GREEN BUTTON

RED BUTTON

GREEN BUTTON

GREEN BUTTON

RED BUTTON

RED BUTTON

GREEN BUTTON

GREEN BUTTON

RED BUTTON

GREEN BUTTON

GREEN BUTTON

RED BUTTON

RED BUTTON

GREEN BUTTON

GREEN BUTTON

RED BUTTONGREEN

BUTTON

GREEN BUTTON

RED BUTTON

RED BUTTON

RED BUTTON

GREEN BUTTON

GREEN BUTTON

RED BUTTON

GREEN BUTTON

GREEN BUTTON

RED BUTTON

RED BUTTON

GREEN BUTTON

GREEN BUTTON

RED BUTTON

GREEN BUTTON

GREEN BUTTON

RED BUTTON

RED BUTTON

RED BUTTON

GREEN BUTTON

GREEN BUTTON

RED BUTTON

GREEN BUTTON

GREEN BUTTON

RED BUTTON

RED BUTTON

GREEN BUTTON

GREEN BUTTON

RED BUTTON

GREEN BUTTON

Tree diagram of all

possibilities when picking 3 red buttons and 5 green ones out of 5

draws.

The purple lines are

representing the path to 10

points or more.

ROUND SIX

P (10 points∨more )=38× 27× 56× 15× 44=0.018=1.8%

ROUND NINE

P (10 points∨more )=58× 47× 36× 25× 14=0.018=1.8%

ROUND TEN

P (10 points∨more )=58× 37× 26× 15× 44=0.018=1.8%

What are the chances that I won in the one round that I won?

ROUND THREE

P (10 points∨more )=58× 37× 46× 25× 14=0.018=1.8%

What are the chances to get 10 points if you have picked __________ button(s)?

1. Green

1. Red

1. Green2. Red

1. Green2. Green

1. Red2. Green

1. Red 2. Red

First, we calculate what are the chances to get 10 points or more at all.

P (10 points∨more )=10×1.8%=18%

a) First picked green button

14 | P a g e

P (greenbutton|10 (¿more ) points )=P (getting10∨more points if greenbuttonis picked first )P (10∨more points )

=

58×4×0.018

0.18=0.0450. 18

=0.25=25%

The chances to win after first picking the green button first is 27.7%

b) First picked red button

P(red button∨10 (¿more ) points)= P (getting10∨more points if red buttonis picked first )P (10∨more points )

=

38×6×0.018

0.18=0.0410.18

=0.227=22.7 %

The chances to win after first picking the red button first is 20.9%

The reason why the chances to win are smaller after first picking out the red button (worth more points) first is because you are just increasing the number of 1-point buttons and decreasing the number of 3-point buttons, creating odds less in you favour.

c) First button green, second button red

P(greenbutton first ,red button second∨10 (¿more ) points)= P (getting 10∨more points if greenbuttonthen red is picked )P (10∨more points )

=

58× 37×3×0.018

0.18=0.0140.18

=0.077=7.7%

The chances to win after picking out the green then the red button is 8.6%

d) First and second buttons green

P( first∧second buttonsgreen∨10 (¿more ) points)=P (getting10∨more points if first∧second buttons picked are green )P (10∨more points )

=

58× 47×1×0.018

0.18= 0.0060.18

=0.037=3.7%

The chances to win after picking out two green buttons is 3.7%

The reason why the chances are smaller to win after picking out two green buttons is because you would have to pick out all the reamaining red ones, and as you pick the red ones, the chances are lower to pick one out again.

e) First button red, second button green

P( first button red , second buttongreen∨10 (¿more ) points)= P (getting 10∨more points if first buttonis red∧second onegreen )P (10∨more points )

=

38× 57×3×0.018

0.18=0.0140.18

=0.086=8.6 %

The chances to win after picking the red button first then the green one is 8.6%

f) Both buttons red

15 | P a g e

P( first∧second buttons red∨10 (¿more ) points)=P (getting10∨more points if first∧second buttons picked are red )P (10∨more points )

=

38× 27×2×0.018

0.18=0.0030.18

=0.018=1.8%

The chances to win after picking two red button in a row is 1.8%

The reason why it is less statistically likely to have won after picking out 2 red buttons is

because you hace just 1 red button and 5 green ones and a 15 chance to pick out the red

button.

4.CONSLUSIONFrom the data which was collected and calculated I have realized that I did not

understand the probability of similar games to this one. I would think if the most valuble object was picked first that my chances were better to win the game overall. But, the opposite has been proven to be true, considering that the chances to win the game after first picking the green button is 27.7% and the chances to win after picking the red button first is 20.9%

16 | P a g e

5.BIBLIOGRAPHY

Haese, Robert, Sandra Haese, Michael Haese, Marjut Maenpaa, and Mark Humphries. Mathematics for the International Student: Mathematics SL. Adelaide: haese Mathematics, 2012. Print.

“Basic statistics”, unknown<www.shmoop.com/basic-statistics-probability/compound-events.html> (28.9.2014)

17 | P a g e

Dependent Events“, Math Goodies, 2014, <http://www.mathgoodies.com/lessons/vol6/dependent_events.html> (28.9.2014)

„Probability - Definition and More from the Free Merriam-Webster Dictionary“, Merriam-Webster, 2014, <http://www.merriam-webster.com/dictionary/probability> (28.9.2014.)

„Conditional probability - Wikipedia, the free encyclopedia“, Wikipedia, <http://en.wikipedia.org/wiki/Conditional_probability> (1.11.2014)

18 | P a g e