Mathematics Project on Probability

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Experiments 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 :- (i) The teacher asked us to either work individually or at most in groups.

(ii)    I collected the following data by visiting say 10 classrooms in the school.

(iii)  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.

(iv)   I made a pie-diagram from the recorded data.

(v)    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.

The actual fraction of children born in January = 38/500 Probability for a child to have birthday in January = 31/365

(vi)    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 by using formula (theoretical values)

31

Probability for a child to have Birthday in January =-- = 0.08

365

29

Probability for a child to have Birthday in February (leap year) =-- = 0.079

365

31

Probability for a child to have Birthday in March =-- = 0.08

365

30

Probability for a child to have Birthday in April =--- = 0.082

365

31

Probability for a child to have Birthday in May =--- = 0.084

365

30

Probability for a child to have Birthday in June =-- = 0.082

365

31

Probability for a child to have Birthday in July =-- = 0.084

365

31

Probability for a child to have Birthday in August =-- = 0.084

365

30

Probability for a child to have Birthday in September =--- = 0.082

365

31

Probability for a child to have Birthday in October =--- = 0.084

365

30

Probability for a child to have Birthday in November =--- = 0.082

365

31

Probability for a child to have Birthday in December =-- = 0.084

365

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

No. of Students

Calculation of probability based on these observations (only 1 class of 61 students)

These are actual experimental values

Probability in month of January

_ favourable outcomes total outcomes

favourable%outcomes total%outcomes

favourable%outcomes total%outcomes

favourable%outcomes total%outcomes

favourable%outcomes total%outcomes

favourable%outcomes total%outcomes

favourable%outcomes total%outcomes

favourable%outcomes total%outcomes

favourable%outcomes total%outcomes

favourable%outcomes total%outcomes

favourable%outcomes total%outcomes

favourable%outcomes total%outcomes

= 12 = 0.19

61

= 0.06 = 0.16 = 0.05 = 0.03

61

1 61

_ 3

61

_ 2

61

= - = 0.16

61

= - = 0.15

61

= - = 0.05

61

= - = 0.09

61

= 0.15 = 0.09 = 0.08

61

6 61

_ _5_

61

Probability in month of February Probability in month of March Probability in month of April Probability in month of May Probability in month of June Probability in month of July Probability in month of August Probability in month of September Probability in month of October Probability in month of November Probability in month of December

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

No. of students

Calculation of probability based on these observations (only 2 classes122 students)

These are actual experimental values

favourable outcomes total outcomes

favourable outcomes total outcomes

favourable outcomes total outcomes

favourable outcomes total outcomes

favourable outcomes total outcomes

favourable outcomes total outcomes

favourable outcomes total outcomes

favourable outcomes total outcomes

favourable outcomes total outcomes

favourable outcomes total outcomes

favourable outcomes total outcomes

= — = 0.09

122

Probability in month of January

= 0.03 = 0.01 = 0.02

122

1 122

3 122

= — = 0.02

122

= — = 0.01

122

= — = 0.07

122

= — = 0.02

122

= 0.05 = 0.07 = 0.05

122

9 122

6 122

= — = 0.04

122

Probability in month of February Probability in month of March Probability in month of April Probability in month of May Probability in month of June Probability in month of July Probability in month of August Probability in month of September Probability in month of October Probability in month of November Probability in month of December

favourable outcomes total outcomes

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
VIIIA 56 4 6 4 5 1 3 2 7 5 6 10 3
VIIIC 57 3 5 7 1 4 5 8 2 6 2 9 5
VIIID 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

. . .... . .. c favourable outcomes                                              74 - ~

Probability in month of January = --------------- = — = 0.12

total outcomes             580

favourable outcomes 39

Probability in month of February = --------------------- = — = 0.06

total outcomes 580

Probability in month of March = favourable outcomes = *L = 0.0*

total outcomes            580

Probability in month of April = favourable outcomes = H. = 0.05

total outcomes   580

Probability in month of May = favourable outcomes = ^ = 0.06

f nf n ? 1 1-vi /l n                       COO

total outcomes 580

. ..... •                                   .. r -i                                                favourable outcomes 42 -

Probability in month of June = -------------------- = — = 0.07

total outcomes 580

Probability in month of July = favourable outcomes = i8 = 0.08

total outcomes   580

46

favourable outcomes

Probability in month of August = ------------------- =----- = 0.07

total outcomes      580

51

favourable outcomes

Probability in month of September = -------------------                       =---- = 0.08

total outcomes                                 580

Probability in month of October = favourable o^comes= = 0.08

total outcomes    580

60

favourable outcomes

Probability in month of November = -------------------  =------ = 0.10

total outcomes       580

58

favourable outcomes

Probability in month of December = ------------------- =----- = 0.1

total outcomes       580

Month No. of children Angle
January 74 46
February 39 24.2
March 48 29.8
April 29 18
May 35 21.7
June 42 26.1
July 48 29.8
August 46 28.6
September 51 31.7
October 50 31
November 60 37.2
December 58 36
Total 580 360

Observation ! - I observed that as the no. of students increases the experimental probability and that calculated with formula become almost equal.


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 :- (i) The teacher asked us to either work individually or at most in groups. (ii) I collected the following data by visiting say 10 classrooms in the school. (iii) 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. (iv) I made a pie-diagram from the recorded data. (v) 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. The actual fraction of children born in January = 38/500 Probability for a child to have birthday in January = 31/365 (vi) 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. Experiments on Probability Calculation of probability by using formula (theoretical values) Probability for a child to have Birthday in January = ଷଵ ଷ଺ହ  = 0.08 Probability for a child to have Birthday in February (leap year) = ଶଽ ଷ଺ହ  = 0.079 Probability for a child to have Birthday in March = ଷଵ ଷ଺ହ  = 0.08 Probability for a child to have Birthday in April = ଷ଴ ଷ଺ହ  = 0.082 Probability for a child to have Birthday in May = ଷଵ ଷ଺ହ  = 0.084 Probability for a child to have Birthday in June = ଷ଴ ଷ଺ହ  = 0.082 Probability for a child to have Birthday in July = ଷଵ ଷ଺ହ  = 0.084 Probability for a child to have Birthday in August = ଷଵ ଷ଺ହ  = 0.084 Probability for a child to have Birthday in September = ଷ଴ ଷ଺ହ  = 0.082 Probability for a child to have Birthday in October = ଷଵ ଷ଺ହ  = 0.084 Probability for a child to have Birthday in November = ଷ଴ ଷ଺ହ  = 0.082 Probability for a child to have Birthday in December = ଷଵ ଷ଺ହ  = 0.084 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 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec No. of Students Calculation of probability based on these observations (only 1 class of 61 students) These are actual experimental values Probability in month of January = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଵଶ ଺ଵ = 0.19 Probability in month of February = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ସ ଺ଵ  = 0.06 Probability in month of March = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଵ ଺ଵ  = 0.16 Probability in month of April = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଷ ଺ଵ  = 0.05 Probability in month of May = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଶ ଺ଵ  = 0.03 Probability in month of June = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଵ ଺ଵ  = 0.16 Probability in month of July = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଽ ଺ଵ  = 0.15 Probability in month of August = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଷ ଺ଵ  = 0.05 Probability in month of September = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଺ ଺ଵ  = 0.09 Probability in month of October = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଽ ଺ଵ  = 0.15 Probability in month of November = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଺ ଺ଵ  = 0.09 Probability in month of December = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ହ ଺ଵ  = 0.08 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 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec No. of students Calculation of probability based on these observations (only 2 classes122 students) These are actual experimental values Probability in month of January = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଵଶ ଵଶଶ = 0.09 Probability in month of February = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ସ ଵଶଶ  = 0.03 Probability in month of March = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଵ ଵଶଶ  = 0.01 Probability in month of April = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଷ ଵଶଶ  = 0.02 Probability in month of May = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଶ ଵଶଶ  = 0.02 Probability in month of June = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଵ ଵଶଶ  = 0.01 Probability in month of July = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଽ ଵଶଶ  = 0.07 Probability in month of August = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଷ ଵଶଶ  = 0.02 Probability in month of September = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଺ ଵଶଶ  = 0.05 Probability in month of October = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଽ ଵଶଶ  = 0.07 Probability in month of November = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଺ ଵଶଶ  = 0.05 Probability in month of December = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ହ ଵଶଶ  = 0.04 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 VIIIA 56 4 6 4 5 1 3 2 7 5 6 10 3 VIIIC 57 3 5 7 1 4 5 8 2 6 2 9 5 VIIID 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 Probability in month of January = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଻ସ ହ଼଴ = 0.12 Probability in month of February = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଷଽ ହ଼଴  = 0.06 Probability in month of March = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ସ଼ ହ଼଴  = 0.08 Probability in month of April = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଶଽ ହ଼଴  = 0.05 Probability in month of May = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଷହ ହ଼଴  = 0.06 Probability in month of June = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ସଶ ହ଼଴  = 0.07 Probability in month of July = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ସ଼ ହ଼଴  = 0.08 Probability in month of August = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ସ଺ ହ଼଴  = 0.07 Probability in month of September = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ହଵ ହ଼଴  = 0.08 Probability in month of October = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ହ଴ ହ଼଴  = 0.08 Probability in month of November = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ଺଴ ହ଼଴  = 0.10 Probability in month of December = ௙௔௩௢௨௥௔௕௟௘௢௨௧௖௢௠௘௦ ௧௢௧௔௟௢௨௧௖௢௠௘௦ = ହ଼ ହ଼଴  = 0.1 Month No. of children Angle January 74 46 February 39 24.2 March 48 29.8 April 29 18 May 35 21.7 June 42 26.1 July 48 29.8 August 46 28.6 September 51 31.7 October 50 31 November 60 37.2 December 58 36 Total 580 360 Observation :- I observed that as the no. of students increases the experimental probability and that calculated with formula become almost equal.

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