• 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
15. 15
15
16. 16
16
17. 17
17
18. 18
18
19. 19
19
20. 20
20
21. 21
21

# Math IA type 2. In this task I will be investigating Probabilities and investigating models based on probabilities in a game of tennis.

Extracts from this document...

Introduction

Nikhil Lodha

Mathematics Internal Assessment Type 2

Mathematical Modelling

Modelling Probabilities in games of tennis.

IB Math HL

Mr. Hoggatt

2nd June 2009

Introduction

In this task I will be investigating Probabilities and investigating models based on probabilities in a game of tennis. I will look to start with a relatively easy and simplistic models where Adam and Ben play each other in Club practice and have a set number of point that they will play. I will then look to find an expected value for the number of points that Adam wins. For this expected value I will calculate a standard deviation to see how much does a randomly selected point vary from the mean. I will then look at Non Extended play games where a maximum of 7 points can be played. I will show that there are 70 ways in which the game can be played. I will do this with the help of the binomial probability distribution formula.

I will also calculate the odds of Adam winning the game and then look to generalize my model so that it does not only apply to only Adam and Ben but to any player.

After making generalized model, I will look at extended games where in theory games could go on forever. Here I will look to use the sum of an infinite geometric series to come up with an appropriate model. I will then use that model to find the odds of Adam winning extended games and then I will look to generalize this model too.

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

Finally, I will evaluate the benefits and limitations models such as these.

Part 1

Middle

because it would be  which would be

The standard deviation of this binomial random variable has the formula

Where n is the number of trials which would be 10 here, since there are 10 points played. p would be the probability of success and q the probability of failure and they would be

Therefore:

Therefore a randomly selected point varies from the mean by 1.491points and the mean is 6.667.  Thus normally Adam will win  points which means that Adam makes between 5.176 to 8.158points and therefore we can conclude that Adam normally wins the practice games as he almost certainty gets more than half the points, which is more than 5 points.

Part 2

Now I will look at Non extended play games where to win a game, the player must win with at least 4 points and by at least 2 points, but to save court time, no game is allowed to go beyind 7 points. This means that if deuce is called and each player has 3 points then the next point determines the winner.

I want to show that the number of different ways in which the game can be played is 70 different ways and I will use primarily the binomial coefficient and lucid logic to show this.

The number of possibilities for the points played can be found by using the binomial coefficient formula which is   .

Therefore in order to find the number of possible games played, I will substitute n with the number of points up for contention [for the sake of calculation, this number is not always the total number of points played because the last point must always be won by the player whose probability of winning is being calculated therefore the n will always be the total points played subtracted by 1.]

Conclusion

Also, players and fans can have a realistic idea of the abilities and expectations. For example

There are however some limitations to the model, the main reason why they are not entirely realistic is that the point probabilities are not always constant in reality because of factors such as changing weather conditions, possible injuries and illnesses.

Also, it has been stated that the ‘2 players, Adam and Ben have played each other often enough to know that Adam wins about twice as many points as Ben does’. Even if we assume that they have practiced against each other and played non-extended games against each other enough to know that their point winning ratio is 2:1, still for extended games such with the possibility of games going on indefinitely, we cannot assume that the their point winning ratio will hold as time goes on, simply because they Adam and Ben could not have played indefinitely, in fact they may not have play each other beyond a couple of deuces. Therefore the fact that the point winning ratio is constant in the model but realistically is changing is a limitation to such probability models.

Another limitation or rather downside to such a model would be that the element of surprise of anticipation of the fans would not exist as the expected outcome would tell us the most likely results and watching the game would no longer be enjoyable, the players would be like robots merely doing their job and playing as math dictates!

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. ## Math Studies I.A

,??-????.=?????,,???.,???.-??. ,??-????. is known as the covariance of X and Y. ,??-?? .= ,-?,??-2.-,,(???)-2.-??.. ,??-?? .is called the standard deviation of X. ,??-?? .= ,-?,??-2.-,(?,??-2.)-??.. ,??-?? .is the standard deviation of Y. r=,?????- ,,???.,???.-??.-,-?,??-2.-,,(???)-2.-??..,-?,??-2.-,(?,??-2.)-??... So, r = ,130013258-,,1707100.(7914.39)-115.-,-51470030000-,,(1707100)-2.-115..,-559261.8111-,,(7914.39)-2.-115... =,12529300.01-161645.4262 �120.7778847.

2. ## Math IA - Logan's Logo

To calculate the value of d here, we make use of the following equation: , Which will work because if you add the highest and lowest y-value points of the curve together, then divide by 2, the result would give you an estimate of the position of the x-axis - if it was merely a graph of .

1. ## MODELLING THE COURSE OF A VIRAL ILLNESS AND ITS TREATMENT

in 1 hour, we would need to divide 200000 by 4, which gives 50000 hour viral particles Died? 0 10,000,000 not yet 1 11,196,827 not yet 2 12,542,877 not yet 3 14,056,756 not yet 4 15,759,389 not yet ... ...

2. ## A logistic model

A peculiar outcome is observed for higher values of the initial growth rate. Show this with an initial growth rate of r=2.9. Explain the phenomenon. One can start by writing two ordered pairs (1?104 , 2.9) , (6 ? 104 , 1)

1. ## Maths IA Type 2 Modelling a Functional Building. The independent variable in ...

2 consecutive heights near the maximum where then chose to see the relationship between a 1 meter increases. And the medium height of the roof was chosen. Table 1. How height of roof affects maximum cuboid dimensions and volume Maximum height of roof (m)

2. ## MATH IA- Filling up the petrol tank ARWA and BAO

Seeing the effect of changing p1 on the money saved by Bao (with respect to Arwa), when p2 and d are kept constant at 0.98\$ and 10km respectively p1 (US\$) Money saved by Bao (with respect to Arwa)(US\$) Formula:(eq1) 1.00 1.64 1.10 3.86 1.20 6.09 1.30 8.31 1.40 10.53 Note:

1. ## Math IA Type 1 Circles. The aim of this task is to investigate ...

âAOP’ has two congruent angles, the following can be used: Φ = 180-2(cos-1 ) Now, the Sine Law can be used to find the length of OP’. = , where a=OP’=x, b=AO=1, A=∠OAP= and B=∠OP’A= = x = x = x = OP’ = Upon doing all of the calculations,

2. ## IB Math Methods SL: Internal Assessment on Gold Medal Heights

This would be attributed to Wessig?s surprise performance at the 1980 event; Wessig covertly used a new unique technique not used before which contributed to his breaking the world record by a wide margin. Given the new data expansion, it would seem appropriate to modify the model so that it fits much more with the extra data points.

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