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G.D. GOENKA WORLD SCHOOL

MATH EXPLORATION

Topic: Probability and Expectation

Candidate Name: Krishna Tayal Candidate Number: 002279-0101 School: G.D. Goenka World School Supervisor: Respected Vinod Arora Sir Subject: Mathematics Level: Standard Level

Maths IA Probability and expectation Rationale From age of 14, I started to play cards games with my friends, it developed an interest to win among my friends. So I thought to calculate the relationship between card games and its probability to know my chances of winning or losing. The interesting fact about cards games is that peoples wants to play more whether they win or lose. So from this ia I will investigate why casino owners are always in profit? I am also going to deal with the usage of probability and expectation in our daily life.

Introduction Probability: how likely some event will happen.

Probability of an event happening=

Number of waysit can happen Total number of outcomes

http://www.mathsisfun.com/definitions/probability.html

The probability always lies between 0 to1, that is

0 ≤ P ( E ) ≤1

Random Variable: random variable are the set of all possible values of event. http://www.mathsisfun.com/data/random-variables.html Probability distribution: Use:Let us consider a game with 2 dice Red colour dice represents John Green colour dice represents Bella

Sample space for 2 dice

IF a player scores less then 4, then the player loses $a. Represented by red colour If a player scores between 5 to 10 then the player loses $b. Represented by yellow colour If a players scores more then 10, then players wins $c. Represented by blue colour

Outcomes in Tabular form

J 1 2 3 4 1 5 6

B 1 1 1 1 1 1

J 1 2 3 4 5 6

B 2 2 2 2 2 2

J 1 2 3 4 5 6

B 3 3 3 3 3 3

J 1 2 3 4 5 6

B 4 4 4 4 4 4

J 1 2 3 4 5 6

B 5 5 5 5 5 5

J 1 2 3 4 5 6

B 6 6 6 6 6 6

Let X be a random variable for winning or losing. X= -a,-b,c Negative means player loosing X -a

P(X)

X.p 6 36

−6 a 36

-b

27 36

−27 b 36

C

3 36

3c 36

Total

1

−6 a 27 b 3 c − + 36 36 36

E(x)=

−6 a 27 b 3 c − + 36 36 36

Case1 E(x)= 0 then the game is fair

Case2 E(X)>0 −6 a 27 b 3 c − + >0 36 36 36 −6 a−27 b +3 c> 0 When expected value of X is greater than 0 then in an average player wins. Case 3 E(X)

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MATH EXPLORATION

Topic: Probability and Expectation

Candidate Name: Krishna Tayal Candidate Number: 002279-0101 School: G.D. Goenka World School Supervisor: Respected Vinod Arora Sir Subject: Mathematics Level: Standard Level

Maths IA Probability and expectation Rationale From age of 14, I started to play cards games with my friends, it developed an interest to win among my friends. So I thought to calculate the relationship between card games and its probability to know my chances of winning or losing. The interesting fact about cards games is that peoples wants to play more whether they win or lose. So from this ia I will investigate why casino owners are always in profit? I am also going to deal with the usage of probability and expectation in our daily life.

Introduction Probability: how likely some event will happen.

Probability of an event happening=

Number of waysit can happen Total number of outcomes

http://www.mathsisfun.com/definitions/probability.html

The probability always lies between 0 to1, that is

0 ≤ P ( E ) ≤1

Random Variable: random variable are the set of all possible values of event. http://www.mathsisfun.com/data/random-variables.html Probability distribution: Use:Let us consider a game with 2 dice Red colour dice represents John Green colour dice represents Bella

Sample space for 2 dice

IF a player scores less then 4, then the player loses $a. Represented by red colour If a player scores between 5 to 10 then the player loses $b. Represented by yellow colour If a players scores more then 10, then players wins $c. Represented by blue colour

Outcomes in Tabular form

J 1 2 3 4 1 5 6

B 1 1 1 1 1 1

J 1 2 3 4 5 6

B 2 2 2 2 2 2

J 1 2 3 4 5 6

B 3 3 3 3 3 3

J 1 2 3 4 5 6

B 4 4 4 4 4 4

J 1 2 3 4 5 6

B 5 5 5 5 5 5

J 1 2 3 4 5 6

B 6 6 6 6 6 6

Let X be a random variable for winning or losing. X= -a,-b,c Negative means player loosing X -a

P(X)

X.p 6 36

−6 a 36

-b

27 36

−27 b 36

C

3 36

3c 36

Total

1

−6 a 27 b 3 c − + 36 36 36

E(x)=

−6 a 27 b 3 c − + 36 36 36

Case1 E(x)= 0 then the game is fair

Case2 E(X)>0 −6 a 27 b 3 c − + >0 36 36 36 −6 a−27 b +3 c> 0 When expected value of X is greater than 0 then in an average player wins. Case 3 E(X)