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Rationale
The world today is depended on the concepts of probability. Everything in here acts according to a plausible chance of defined or undefined outcomes. English A Language and literature is one of my three standard level subjects that I have been studying in this course of the IB Diploma Programme. 15% of the grade weighting in this subject depends upon an Individual Oral Commentary. The task requires students to comment or rather scrutinise an extract from a literary text studied in part 4 of the course. In my school, the English facilitator chooses one extract for each student out of six from two literary works. To stay ahead of what would turn out to be my extract, I would try to guess the probability of getting either of the texts. I collect data from students who have already completed their tasks and analyse it. The probability that I would get the first literary text is a 0.5 if an extract is selected randomly. While, the probability that two students consequently get the extract from the same text one is a 0.25. When would occur such a situation when the same text extract is given in succession?

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Even the chart of local trains in the city of Mumbai, though specified, defers to some great extents. For instance, a train headed for “Church Gate” is supposed to arrive in the next 10 minutes after the previous one. However, sometimes the waiting time increases and in worst case even triples. This random behaviour also takes place in winters in Bhilwara, a city in Rajasthan. While some nights the temperature falls down to 8 degree Celsius, there are also times when it reaches as high as 19 degree Celsius. 

In my coursework of Mathematics SL, I have learnt about some probability distributions that explain such situations. While the binomial distribution accounts ‘n’ trials of the number of successes, the negative binomial distribution accounts trials until the ‘r’ number of success. 

In my investigation, I will be exploring the distribution of the situation wherein 2 consecutive successes occur for the first time in ‘n’ number of trials. I also think that this might have a connection with the negative binomial distribution so I will term this as Successive Negative Binomial Distribution “SNBD”. 

If x = number of trials less than and equal to the first situation with 2 consecutive successes and p = the success probability, then Successive Negative Binomial Distribution ~ x. The second parameter should also be considered as there another distribution with n consecutive successes. 

I will further explore this kind of distribution in the most uncomplicated case where in the first success occurs of  2 consecutive Tails when a fair coin is tossed several times. I will also try and find the main features which are: Mode and median, probability distribution function, generation of a function, value and variance expectation. I will account few elements of the general probability case p and also compare the distribution with the ones I have studied in the SL coursework. 

I could put this investigation to use to determine my chances of getting the extract from the same literary text as the student before me during my English Internal Oral Commentary, By this, I could enter the test hall well prepared and certain of my success!

Factual Affirmation
To avoid physical errors, instead of tossing an actual coin, I use the random number generator function in my Graphic Display Calculator (HP Prime) in which an imaginary coin is tossed and the data is recorded simultaneously till the result fo two consecutive tails is achieved. Although, the GDC only generates the number. Thus, I represent 1 = Heads and 2 = Tails. This gives me an insight of some important features of the distribution which I will compare after exploring the theoretical aspect. 

Calculator Outcome:-
Mean: 5.28
Median: 4
Standard Deviation: 3.30
Lower Quartile: 3
Upper Quartile: 6
Interquartile range: 3 

Trial

Trial

Outcome

No. Of tosses

Trial

Outcome

No. Of tosses

1

THHTT

5

26

HTT

3

2

THTT

4

27

HHHTT

5

3

HHHTT

5

28

HHTT

4

4

TT

2

29

THTHTHHHHHTT

12

5

HTT

3

30

HTHTHTT

7

6

THTHHTHHTT

10

31

THTT

4

7

HHTT

4

32

THTHTT

6

8

HHHTT

3

33

THTT

4

9

HTT

5

34

HTT

3

10

TT

2

35

HTHTHTHTHTT

11

11

HTHHTT

6

36

HTHTT

5

12

THHHTT

6

37

TT

2

13

THHHHHHHHTT

11

38

THTT

4

14

HHTT

2

39

TT

2

15

TT

4

40

HHTHTHHTHTT

11

16

HTT

3

41

HTT

3

17

HHTHTT

6

42

HTT

3

18

THHHHTHHTT

10

43

HTHHHHHTHHHHHHHHTT

18

19

TT

2

44

HHTT

4

20

THTHTHTT

8

45

HHHTHTHHTT

10

21

HHTHTT

6

46

HTT

3

22

HTHHHTT

7

47

THTT

4

23

THTHHTT

7

48

HTT

3

24

TT

2

49

HTT

3

25

TT

2

50

THHTT

5

Theory
The Discrete Random Variable x is the number of tosses made till the first 2 consecutive Tails were obtained including the same. 
x ~ SNBD (2, 0.5)
The domain would be; X = 2,3,4,5…. X could go on to tend to infinity with geometric distribution and negative binomial distribution, in theory. 

P(x = 2) = P(TT) = 0.5 x 0.5 = 0.25
P(x = 3) = P(HTT) = 0.5 x 0.5 x 0.5 = 0.125