• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11
12. 12
12
13. 13
13
14. 14
14

# Modelling Probabilities on games of tennis

Extracts from this document...

Introduction

Rushabh K Kamdar

Math HL

MATH PORTFOLIO

Modelling Probabilities on games of tennis

Name of Candidate: Rushabh K Kamdar

Level: Math HL

28th October 2012

Introduction:

In this portfolio I shall investigate the different models and probabilities based on the probabilities in the game of tennis. First I will start with the Part 1 of the portfolio where I will be concluding with the expected value and the standard distribution from my results.

I will then take a look at the Non Extended play games where the highest of 7 points can be played. This is will be done with the use of binomial distribution. Then I will calculate the odds of Adam winning the game of tennis and will generalize my model so that I can apply to any other player. After making this model, I will take a look at the extended games where in theory the game could go on forever. This is the stage where I find a model to find the odds of Adam winning the extended games and then will generalize this also.

In the Part 3, I will also test the model for different values of point winning probabilities and find out the odds for each of them and then will then look for patterns from the values of odds that I find.

Middle

= 6: 2

2 – 8 = 20

y = 7: 2

2 – 8 – 20 = 40

Therefore from this it can be seen that there are (2 + 8 + 20 + 40) = 70 different possible ways in which the game can be played showing the different combinations possible when a player loses all the games. After finding the different ways the game can be played in, now I shall work out the probability of Adam winning the game. The different possible ways in which Adam can win the game are 4-0, 4-1, 4-2 and 4-3; and so now I will have to find the sum of probability of these events taking place.

The model for the probability of a particular outcome of Adam winning the game would be:

P (X=3) =

The part of the equation inside the brackets is the last point and this part of the model has no variables because it is the last point and should always be won by Adam.

Below are the ways in which Adam can win:

 Probability of winning 4-0P (X=3) = =  = Probability of winning 4-1P (X=3) = = 4 = Probability of winning 4-2P (X=3) = = 4 = Probability of winning 4-3P (X=3) = = 4 =

Therefore the probability of Adam winning the game is the sum of the above probabilities:

Hence the total probability =

=

The formula for the odds of Adam winning=

Conclusion

There are also several limitations to my models due to which they are not entirely realistic. The point probabilities, which were given, are never constant in a real game and are influenced by many factors such as the climatic conditions, stamina of the players, injuries, etc., due to which the probability values can sometimes go entirely wrong. Even when we look at the extended game, the possibility of the game going on forever decreases the reliability on the point probabilities. But if we look on the other hand, the models that I have developed shows that how the player’s dominance over the game affect the overall results and also how the variations in the game, like changing the rules increase the possibility of a stronger player to win the game. Therefore this model is also useful to a large extent.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related International Baccalaureate Maths essays

1. ## Extended Essay- Math

ï¿½_ï¿½ï¿½wpFï¿½0ï¿½{bMï¿½xï¿½>%ï¿½ï¿½oï¿½oï¿½ï¿½ï¿½a8ï¿½jjï¿½ &Sï¿½,vï¿½bFl.iï¿½ï¿½2ï¿½ ï¿½@ï¿½ i#ï¿½ ï¿½Fï¿½ï¿½ï¿½V ï¿½"ï¿½ï¿½ÅPï¿½ï¿½ï¿½ÈHdL\$ï¿½ï¿½ï¿½ï¿½3ï¿½_ï¿½_ãqï¿½)'ï¿½!1/4ï¿½ï¿½{6~dï¿½?2(c)ï¿½ï¿½ï¿½dï¿½ï¿½3/4ï¿½b1/2(c)ï¿½ï¿½ï¿½Ç1/4_ï¿½ï¿½_ï¿½ï¿½-ï¿½ï¿½Zï¿½ï¿½VAï¿½uÐºh ï¿½ï¿½fEsï¿½2Zmï¿½ï¿½Ck!}ï¿½1/4G\$ ï¿½ï¿½qG~ï¿½ï¿½ï¿½"ï¿½ï¿½dM-`ï¿½wï¿½ï¿½ï¿½.ï¿½ï¿½(c)ï¿½<ï¿½/ uxï¿½}ï¿½ï¿½\$!ï¿½ï¿½È¨ï¿½jPpï¿½ï¿½rï¿½ï¿½-"ï¿½dï¿½ï¿½ï¿½vï¿½ï¿½ï¿½\;1ë·²_~ï¿½"ï¿½ï¿½Ñ¿h!(c)ï¿½ï¿½!ï¿½xï¿½_ï¿½ï¿½qï¿½^!ï¿½"ï¿½_4ï¿½ï¿½;ï¿½ï¿½ï¿½ï¿½"ï¿½[-ï¿½\$ï¿½#-ï¿½xï¿½GÞ³"[email protected]ï¿½Kï¿½ 8wï¿½ ï¿½`ï¿½ [email protected]"ï¿½ ï¿½@.ï¿½ ï¿½ PNï¿½Pï¿½ï¿½ï¿½ nï¿½^p <#ï¿½)ï¿½S`|ï¿½ï¿½;Xï¿½ ' &ï¿½ï¿½D )HRï¿½t!ï¿½rï¿½ï¿½!(ï¿½ï¿½ï¿½1/2ï¿½>(*"* ï¿½Ptï¿½"zï¿½{ï¿½cï¿½9ï¿½ï¿½ï¿½-ï¿½ï¿½0 ï¿½...(tm)a-X-ï¿½ï¿½aï¿½ vï¿½1/2ï¿½ 8Nï¿½sï¿½ï¿½ (r)ï¿½ï¿½ï¿½6ï¿½~?...ï¿½ï¿½Oï¿½ ï¿½(V"Jï¿½ï¿½2Bï¿½By Q1ï¿½tï¿½!T ï¿½uï¿½...ï¿½ï¿½CMï¿½Pkh,ï¿½ Í-Aï¿½ï¿½ï¿½CGï¿½ï¿½Ñï¿½ï¿½ztï¿½=ï¿½~ï¿½^DoaHnï¿½Fcï¿½qï¿½a1ï¿½ï¿½Lï¿½3ï¿½yï¿½(tm)ï¿½|ï¿½bï¿½ï¿½X1ï¿½ï¿½ï¿½ ï¿½ï¿½bc"ï¿½-ï¿½-ï¿½cï¿½4vï¿½ï¿½qï¿½p:ï¿½]8 . -ï¿½+ï¿½5ï¿½nï¿½Fq3ï¿½x"ï¿½ï¿½ï¿½7ï¿½{ï¿½#ï¿½ï¿½|#3/4?ï¿½ï¿½Å¯"Mï¿½.ï¿½?!(tm)p"PKï¿½"<"ï¿½ï¿½iiï¿½hthï¿½hBhï¿½hï¿½h.ï¿½ ï¿½1/4ï¿½ï¿½J\$ï¿½D{"ï¿½ï¿½I,#^" ï¿½ï¿½hÉ´'ï¿½Fï¿½ï¿½ï¿½ï¿½ï¿½ï¿½i{hï¿½ï¿½~%'Hï¿½\$}' )ï¿½[email protected] ï¿½&1/2&ï¿½ cï¿½"ï¿½ï¿½ ï¿½ï¿½ "ï¿½kï¿½ï¿½ï¿½BO ï¿½7 ï¿½Cï¿½B_Bï¿½ï¿½ï¿½ï¿½A"ï¿½ï¿½ï¿½Âï¿½Pï¿½ï¿½ï¿½ï¿½a...'ï¿½Qï¿½qc8ï¿½aï¿½Fï¿½{ï¿½sdY"lBï¿½'ï¿½kÈ·ï¿½ï¿½L(&!&#&?ï¿½}Lï¿½LL3ï¿½Xf1f ï¿½ï¿½|ï¿½fï¿½aï¿½E2ï¿½2ï¿½ KK%ï¿½M-)V"("kï¿½Qï¿½+ï¿½ï¿½?ï¿½xï¿½ ï¿½ï¿½ï¿½ï¿½.ï¿½ï¿½ï¿½ï¿½ï¿½sï¿½ï¿½ ï¿½boaï¿½"ï¿½ï¿½ï¿½"#"ï¿½8G;ï¿½+N4ï¿½\$ï¿½=g"ï¿½)ï¿½ï¿½.f.-.?(r)C\Wï¿½^pï¿½ï¿½'ï¿½ ï¿½(c)ï¿½5ï¿½Cï¿½+<1/4<f<Q<ï¿½<ï¿½yxYyï¿½yCxï¿½y"yï¿½ï¿½tï¿½ï¿½|ï¿½|ï¿½ï¿½>ï¿½ï¿½ï¿½ï¿½-ï¿½ï¿½/ p ï¿½ ï¿½ ï¿½Xtï¿½l|%D#ï¿½.(T,ï¿½'ï¿½(ï¿½'l#1/4Wï¿½ï¿½ï¿½ ï¿½ï¿½ï¿½Hï¿½H(c)ï¿½]'UQ1QWï¿½ï¿½ï¿½sbï¿½bb)bï¿½^ï¿½"ï¿½Ä£ï¿½"ÅH`%ï¿½%B%ï¿½\$F\$aIï¿½`ï¿½Jï¿½GRï¿½"ï¿½Uï¿½Jï¿½4FZC:Bï¿½Zï¿½(tm)

2. ## A logistic model

The curve approaches (and becomes) the value of 4.22x104 fish, which is the annual stable fish population with an annual harvest of 7.5?103 fish. b. Consider a harvest of 1? 104 fish. Then the logistic function model is: ?5 2 4 un?1 ? (?1? 10 )(un ) ? 1.6(un )

1. ## THE DICE GAME - calculating probabilities

Bob-3 Ann-1, Bob-4 Ann-2, Bob-4 Ann-3, Bob-4 Ann-4, Bob-4 Ann-5, Bob-4 Ann-6, Bob-4 Ann-1, Bob-5 Ann-2, Bob-5 Ann-3, Bob-5 Ann-4, Bob-5 Ann-5, Bob-5 Ann-6, Bob-5 Ann-1, Bob-6 Ann-2, Bob-6 Ann-3, Bob-6 Ann-4, Bob-6 Ann-5, Bob-6 Ann-6, Bob-6 (The bold combinations are cases in which Ann wins.)

2. ## This essay will examine theoretical and experimental probability in relation to the Korean card ...

Understanding the Concepts 1. Probability 1. What is Probability? There are number of different definitions of mathematical probability which have been proposed by various authors. The scientific definition of a 'fundamental' concept such as "the concept of probability is merely the refinement and logical processing of a series of very

1. ## Math IA type 2. In this task I will be investigating Probabilities and investigating ...

Adam and Ben and we need to find how many points each player wins so r is substituted with A as a variable where it represents the number of points which are won by Adam and therefore would represent the number of points won by Ben.

2. ## Creating a logistic model

value of a, and a should always have a value of 5, the calculations by the calculator are not the same. So why is this not the case for the logistic functions calculated by the GDC? The model that we used to insert data values for the population of fish is only an estimate.

1. ## Modelling Probabilities in Games of Tennis

and P(B). Possible alterations could be the conditions the players play under. Adam could play a lot worst when the temperature increases, whereas Ben might play better on hard courts than on grass courts. Chance simply represents either Ben or Adam having a good day.

2. ## The purpose of this investigation is to create and model a dice-based casino game ...

Let the ordered pair (m,n) represent the case in which player A rolls a number m and player B rolls a number n such that m and n are integers, , and . Now consider the ordered pair (a,b) where . For every such pair (a,b), there exists another pair (b,a); thus there exists a one-to-one correspondence between the sets of (a,b)

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to