Mathematics Project on Probability

Objective: To appreciate that finding probability through experiment is different from finding probability by calculation. Students become sensitive towards the fact that if they increase the number of observations, probability found through experiment approaches the calculated probability.

Description:

The teacher asked us to either work individually or at most in groups.
I collected the following data by visiting say 10 classrooms in the school.
I obtained the fraction of number of children having their birthday in the month of January, February, … December from the data given in the table.
I made a pie-diagram from the recorded data.
I investigated that fraction actually obtained in step 3 tallies with the calculated probability obtained for each month.

If total no. of children whose birthday falls in month of January is 38 and the total number of students is 500:

Actual fraction of children born in January = ³⁸⁄₅₀₀
Probability for a child to have birthday in January = ³¹⁄₃₆₅

The students increased the sample size, i.e. increased the number of observations and studied if the actual fraction approaches the calculated probability. I used a random sample for this purpose.

Calculation of Probability Using Formula (Theoretical Values)

Probability for a child to have Birthday in January = ³¹⁄₃₆₅ = 0.08
Probability for a child to have Birthday in February (leap year) = ²⁹⁄₃₆₅ = 0.079
Probability for a child to have Birthday in March = ³¹⁄₃₆₅ = 0.08
Probability for a child to have Birthday in April = ³⁰⁄₃₆₅ = 0.082
Probability for a child to have Birthday in May = ³¹⁄₃₆₅ = 0.084
Probability for a child to have Birthday in June = ³⁰⁄₃₆₅ = 0.082
Probability for a child to have Birthday in July = ³¹⁄₃₆₅ = 0.084
Probability for a child to have Birthday in August = ³¹⁄₃₆₅ = 0.084
Probability for a child to have Birthday in September = ³⁰⁄₃₆₅ = 0.082
Probability for a child to have Birthday in October = ³¹⁄₃₆₅ = 0.084
Probability for a child to have Birthday in November = ³⁰⁄₃₆₅ = 0.082
Probability for a child to have Birthday in December = ³¹⁄₃₆₅ = 0.084

Experimental Observations

Table 1: Number of Students Having Birthday in Various Months in 1 Class

Class	No. of Students	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
VI C	61	12	4	1	3	2	1	9	3	6	9	6	5

Calculation of Probability Based on 1 Class (61 Students)

These are actual experimental values (favourable outcomes / total outcomes):

Probability in month of January = ¹²⁄₆₁ = 0.19
Probability in month of February = ⁴⁄₆₁ = 0.06
Probability in month of March = ¹⁄₆₁ = 0.016
Probability in month of April = ³⁄₆₁ = 0.05
Probability in month of May = ²⁄₆₁ = 0.03
Probability in month of June = ¹⁄₆₁ = 0.016
Probability in month of July = ⁹⁄₆₁ = 0.16
Probability in month of August = ³⁄₆₁ = 0.05
Probability in month of September = ⁶⁄₆₁ = 0.09
Probability in month of October = ⁹⁄₆₁ = 0.15
Probability in month of November = ⁶⁄₆₁ = 0.09
Probability in month of December = ⁵⁄₆₁ = 0.08

Table 2: Number of Students Having Birthday in Various Months in 2 Classes

Class	No. of Students	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
VI C	61	12	4	1	3	2	1	9	3	6	9	6	5
VI D	61	9	2	2	1	5	2	6	8	7	3	10	6

Calculation of Probability Based on 2 Classes (122 Students)

These are actual experimental values (favourable outcomes / total outcomes):

Probability in month of January = ²¹⁄₁₂₂ = 0.17
Probability in month of February = ⁶⁄₁₂₂ = 0.05
Probability in month of March = ³⁄₁₂₂ = 0.02
Probability in month of April = ⁴⁄₁₂₂ = 0.03
Probability in month of May = ⁷⁄₁₂₂ = 0.06
Probability in month of June = ³⁄₁₂₂ = 0.02
Probability in month of July = ¹⁵⁄₁₂₂ = 0.12
Probability in month of August = ¹¹⁄₁₂₂ = 0.09
Probability in month of September = ¹³⁄₁₂₂ = 0.11
Probability in month of October = ¹²⁄₁₂₂ = 0.10
Probability in month of November = ¹⁶⁄₁₂₂ = 0.13
Probability in month of December = ¹¹⁄₁₂₂ = 0.09

Table 3: Number of Students Having Birthday in Various Months in 10 Classes

Class	No. of Students	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
VI C	61	12	4	1	3	2	1	9	3	6	9	6	5
VI D	61	9	2	2	1	5	2	6	8	7	3	10	6
VII B	59	3	4	2	3	4	5	4	5	5	10	8	5
VIII A	56	4	6	4	5	1	3	2	7	5	6	10	3
VIII C	57	3	5	7	1	4	5	8	2	6	2	9	5
VIII D	61	8	2	10	3	8	5	1	6	4	6	4	4
IX B	60	6	2	5	1	4	6	6	4	3	5	5	13
IX D	58	12	6	7	2	3	5	1	4	2	4	5	7
X B	53	9	7	6	4	1	3	3	5	6	4	1	4
X D	54	8	1	4	6	3	7	8	2	7	1	2	5
Total	580	74	39	48	29	34	42	48	46	51	50	60	58

Calculation of Probability Based on 10 Classes (580 Students)

Probability in month of January = ⁷⁴⁄₅₈₀ = 0.12
Probability in month of February = ³⁹⁄₅₈₀ = 0.06
Probability in month of March = ⁴⁸⁄₅₈₀ = 0.08
Probability in month of April = ²⁹⁄₅₈₀ = 0.05
Probability in month of May = ³⁴⁄₅₈₀ = 0.06
Probability in month of June = ⁴²⁄₅₈₀ = 0.07
Probability in month of July = ⁴⁸⁄₅₈₀ = 0.08
Probability in month of August = ⁴⁶⁄₅₈₀ = 0.07
Probability in month of September = ⁵¹⁄₅₈₀ = 0.08
Probability in month of October = ⁵⁰⁄₅₈₀ = 0.08
Probability in month of November = ⁶⁰⁄₅₈₀ = 0.10
Probability in month of December = ⁵⁸⁄₅₈₀ = 0.10

Pie Diagram Data

Month	No. of Children	Angle (degrees)
January	74	46.0
February	39	24.2
March	48	29.8
April	29	18.0
May	35	21.7
June	42	26.1
July	48	29.8
August	46	28.6
September	51	31.7
October	50	31.0
November	60	37.2
December	58	36.0
Total	580	360.0

Graph: Pie diagram showing the distribution of student birthdays across all 12 months, with sector angles proportional to the number of students born in each month (plotted from data above).

Observation

I observed that as the number of students increases, the experimental probability and that calculated with the formula become almost equal.

Mathematics Project on Probability