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PROBABILITY OF CLASS 10TH
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CL-X MTA CH-4 PART-3.pdf
1.
ଶ୍ରେଣୀ- ଦଶଭ ଚତୁ ଥଥ
ଅଧ୍ୟାୟ (ସମ୍ଭାଫୟତା ) (ବାଗ - 3 ) ଶିକ୍ଷକ – ଫିଶ୍ରଯନ୍ଦ୍ର କୁଭାଯ ଆଚାମଥୟ ସଯକାଯୀ ଉନ୍ନୀତ ଉଚ୍ଚ ଫିଦୟାୟ ତଶ୍ରଫଗା, ଜିଲ୍ଲା - କାହାଣ୍ଡି
2.
ଆଶ୍ରଭ ଢିଥିଶ୍ରର 1. ଅନୁବ ୂ
ଭିକ ସମ୍ଭାଫୟତା ଫା Emperical Probability “ଘଟଣାଯ ଆନୁବଫିକ ସମ୍ଭାଫୟତା” = ଆବଶ୍ୟକ ଫଲର ବାରମବାରତା ସମୁଦାୟ ପରୀକ୍ଷଣ ସଂଖ୍ୟା 2. ତତ୍ତ୍ଵାଧାଯିକ ସମ୍ଭାଫୟତା ଫା Theoritical ଫା Classical Probability “ଘଟଣା E ଯ ସମ୍ଭାଫୟତା “ P(E) = ଅନୁଗୃହିତ ଫଲ ସଂଖ୍ୟା ପରିକ୍ଷଣର ସମ୍ଭାବୟ ସମସ୍ତ ଫଲାଫଲ ସଂଖ୍ୟା ଏହା ଯୀକ୍ଷଣ ସିଦ୍ଧ
3.
ଶ୍ରସଟ ତତ୍ତ୍ଵ ସହାୟତାଶ୍ରଯ
ସମ୍ଭାଫୟତାଯ ଧାଯଣା SAMPLE SPACE (S) : ଶ୍ରକୌଣସି ଯୀକ୍ଷଣଯ ସଭସ୍ତ ପାପ କୁ ଶ୍ରନଇ ଗଠିତ ଶ୍ରସଟ କୁ sample space କୁହାମାଏ I ଉଦାହଯଣ: ଶ୍ରଗାଟିଏ ଭୁଦ୍ରାକୁ ଥଶ୍ରଯ ଟସ କଶ୍ରର, S= { H,T} ଏଠାଶ୍ରଯ ISI=2 ଶ୍ରଗାଟିଏ ରୁ ଡ ୁ ଶ୍ରଗାଟିଏ 1 ଥଯ ଟସ କଶ୍ରର, S= {1,2,3,4,5,6}, ISI = 6 ଇତୟାଦି I
4.
ଭୁଦ୍ରା ଟସ ଶ୍ରକ୍ଷତ୍ରଶ୍ରଯ
“s” ନିର୍ଣ୍ଥୟ 1ଥଯ ଟସ 2ଥଯ ଟସ 3ଥଯ ଟସ H HH HHH HHT HT HTH HTT T TH THH THT TT TTH TTT
5.
ସିଦ୍ଧାନ୍ତ : ଶ୍ରଗାଟିଏ
ଅପ୍ରଫଣ ଭୁଦ୍ରାକୁ n ଥଯ ଟସ କଶ୍ରର sample space ‘s’ ଯ ଉାଦାନ ସଂଖ୍ୟା ISI = 𝟐𝒏 Example: ଶ୍ରଗାଟିଏ ଅପ୍ରଫଣ ଭୁଦ୍ରାକୁ (i) 6 ଥଯ ଟସ କଶ୍ରର ISI = 𝟐𝟔 = 64 (ii) 6 ଥଯ ଟସ କଶ୍ରର ISI = 𝟐10 = 1032
6.
11 12 13
14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 56 61 62 63 64 65 66 ରୁ ଡ ୁ ଶ୍ରଗାଟି 2 ଥଯ ଗଡାଇଶ୍ରର
7.
ରୁ ଡ଼ୁଶ୍ରଗାଟିକୁ ଟସ
ଶ୍ରକ୍ଷତ୍ରଶ୍ରଯ sample space ଯ ଉାଦାନ ସଂଖ୍ୟା ISI ନିର୍ଣ୍ଥୟ I (i) ରୁ ଡ ୁ ଶ୍ରଗାଟି 1 ଥଯ ଗଡାଇଶ୍ରର s = {1,2,3,4,5,6}, ISI = 6 = 𝟔𝟏 (ii) ରୁ ଡ ୁ ଶ୍ରଗାଟି 2 ଥଯ ଗଡାଇଶ୍ରର s = { 11,22,33,44,55,66, 21,22,23,24,25,26, 31,32,33,34,35,36, 41,42,43,44,45,46, 51,52,53,54,55,56, 61,62,63,64,65,66 } ISI = 36 = 𝟔𝟐
8.
ସିଦ୍ଧାନ୍ତ : ଶ୍ରଗାଟିଏ
ରୁ ଡ ୁ ଶ୍ରଗାଟିକୁ n ଥଯ ଗଡାଇଶ୍ରର sample space ‘S’ ଯ ଉାଦାନ ସଂଖ୍ୟା ISI = 𝟔𝒏 ଉଦାହଯଣ : ରୁ ଡ ୁ ଶ୍ରଗାଟି ଶ୍ରକ୍ଷତ୍ରଶ୍ରଯ (i) 3 ଥଯ ଗଡାଇଶ୍ରର ISI = 𝟔𝟑 = 216 (ii) 4 ଥଯ ଗଡାଇଶ୍ରର ISI = 𝟔𝟒 = 1296
9.
ଏକ ଘଟଣା E
ଶ୍ରହଶ୍ରର, ଯୀକ୍ଷଣଯ ଅଫଶିଷ୍ଟ ଘଟଣା ଗୁଡିକୁ E ଯ ଯିୂଯକ ଘଟଣା ଫା (COMPLEMENTARY EVENT) କୁହାମାଏ I E ଯ ଯିୂଯକ ଘଟଣାକୁ E ଫା E’ ଯ ୂ ଶ୍ର ଶ୍ରରଖ୍ାମାଏ I P(E) + P(E’) = 1 ଫା P(E’) = 1 – P(E)
10.
ସଯ ଫା ଶ୍ରଭୌିକ
ଘଟଣା ( Simple or Elementary Event) ଏକ ଉାଦାନ ଫିଶିଷ୍ଟ ଘଟଣାକୁ ସଯ ଫା ଶ୍ରଭୌିକ ଘଟଣା କୁହାମାଏ I ଉଦାହଯଣ : ଭୁଦ୍ରାକୁ 1 ଥଯ ଟସ କଶ୍ରର E1 = {H} , E2 = {T} E1 ଓ E2 ପ୍ରଶ୍ରତୟାକ ଘଟଣା I ଭୁଦ୍ରାକୁ 2 ଥଯ ଟସ କଶ୍ରର E1 = {HH} , E2 = {HT}, E3 = {TH} , E4 = {TT} E1,E2 , E3 ଓ E4 ପ୍ରଶ୍ରତୟାକ ଶ୍ରଗାଟିଏ ଶ୍ରଗାଟିଏ ଶ୍ରଭୌିକ ଘଟଣା I
11.
ଶ୍ରମୌଗିକ ଘଟଣା (Compound
Eevent) ଏକାଧିକ ଉପାଦାନ ବିଶ୍ିଷ୍ଟ ଘଟଣା କୁ ୄଯାଗିକ ଘଟଣା କୁହଯାଏ। Ex- ୄଗାଟାଏ ମୁଦ୍ରା କୁ 2 ଥର ଟସ କରାଗା E1=ଅତି କମ ୄର ୄଗାଟାଏ H ଥିବା ={HH, HT, TH} ଏଠାୄର [E1]= 3 ଏଣୁ E1 ଏକ ୄଯୌଗିକ ଘଟଣା E2=ୄକବଲ H ବା ୄକବଲ T ଥିବା = {HH, TT} ଏଠାୄର IE2I=2 ଏହା ମଧ୍ୟ ୄଯୌଗିକ ଘଟଣା
12.
ଯସ୍ପଯ ଫହିବଥ ୂ କ୍ତ
ଘଟଣା (Mutually Execlusive Events) ଦୁଇଟି ଘଟଣା E1 ଓ E2 (E1 C S, E2 C S) ଯସ୍ପଯ ଫହିବଥ ୂ କ୍ତ ଶ୍ରହଫ ମଦି E1 ∩ E2= ∅ ଶ୍ରହଫ Ex- ଭୁଦ୍ରାକୁ 2 ଥଯ ଟସ କଶ୍ର E1: ଶ୍ରକଫ H ଆସିଫ = > E1= {HH} E2: ଶ୍ରକଫ T ଆସିଫ => E2= {TT} => E1 ∩ E2= ∅
13.
ଯିୂଯକ ଘଟଣା (Complementary
Events) E1 ଓ E2 ଘଟଣା ଦ୍ୱୟ ପରିପୂରକ ୄହବ ଯଦି E1 ଓ E2 ପରସ୍ପର ବହିଭଭ ୂ କ୍ତ ଓ ୄସମାନଙ୍କ ସଂୄଯାଗ ୄହତୁ S ସୃଷ୍ଟି ୄହବ । Ex: ୄଗାଟାଏ ୁଡ ୁ ୄଗାଟି 1 ଥର ୄଗାଡାଇୄ S= {1,2,3,4,5,6} E1= ଫ ଏକ ଅଯୁଗମ ସଂଖ୍ୟା = {1,3,5} E2= ଫ ଏକ ଯୁଗମ ସଂଖ୍ୟା = {2,4,6} ଏଠାୄର E1 ∩ E2= ∅ ଓ E1 ∪ E2= S ୄତଣୁ E1 ଓ E2 ପରସ୍ପର ପରିପୂରକ ଘଟଣା। E ଘଟଣା ପରିପୂରକ ଘଟଣା କୁ E’ ବା E ର ୂ ୄପ ୄଖ୍ାଯାଇଥାଏ [ସମସ୍ତ ବହିିଃଭ ୁ କ୍ତ ଘଟଣା ପରିପୂରକ ନୁହନ୍ତି କିନ୍ତୁ ସମସ୍ତ ପରିପୂରକ ଘଟଣା ବହିିଃଭ ୁ କ୍ତ ଅୄଟ I ]
14.
ଏକ ଘଟଣା E
ଯ ସମ୍ଭାଫୟତା P(E) = 𝑰𝑬𝑰 𝑰𝑺𝑰 ସମ୍ଭାଫୟତା ଯ ଶ୍ରକଶ୍ରତକ ଧଭଥ : I. E C S ଶ୍ରହଶ୍ରର P(∅) = 𝟎, 0≤ P(E) ≤ 1, P(S) = 1 ଏଠାଶ୍ରଯ ∅ = ଅନିଶ୍ଚିତ ଘଟଣା (impossible event) S= ନିଶ୍ଚିତ ଘଟଣା (SURE EVENT )
15.
ଏକ ଘଟଣା (E)
ଓ ଏହାଯ ଯିୂଯକ ଘଟଣା E’ ଦ୍ୱୟ S ଯ ଉଶ୍ରସଟ ଏଫଂ P(E) + P(E’) = 1
16.
E1 ଓ E2
ଦୁଇଟି ଘଟଣା ଓ E1 C S, E2 C S ଶ୍ରହଶ୍ରର E1 ∪ E2 , E1 ∩ E2, ଭଧ୍ୟ ଶ୍ରଗାଟିଏ ଶ୍ରଗାଟିଏ ଘଟଣା ଏଫଂ (E1 ∪ E2 ) C S, (E1 ∩ E2) C S, ଶ୍ରସଟ ତତ୍ଵ ଆନୁମାୟୀ I E1 ∪ E2 I = IE1I +IE2I - I E1 ∩ E2I ଶ୍ରତଣୁ P (E1 ∪ E2 ) = I E1 ∪ E2 I 𝑰𝑺𝑰 => P (E1 ∪ E2 ) = IE1I +IE2I − I E1 ∩ E2I 𝑰𝑺𝑰 P (E1 ∪ E2 ) = IE1I 𝑰𝑺𝑰 + IE𝟐I 𝑰𝑺𝑰 - I E1 ∩ E2I 𝑰𝑺𝑰 E1 ଓ E2 ଯସ୍ପଯ ଫହିିଃବ ୂ କ୍ତ ଶ୍ରହଶ୍ରର , E1 ∩ E2= ∅ => P(E1 ∩ E2)=0, => P (E1 ∪ E2 ) = P(E1) + P(E2)
17.
ୄଗାଟିଏ ୁ ଡ ୁ
ୄଗାଟିକୁ 1 ଥର ଗଡାଇୄ ଫଲଟି “ ଏକ ଯୁଗମ ସଂଖ୍ୟା କିମବା ଫଲ ≥ 4" ୄହବାର ସମ୍ଭାବୟତା ସ୍ଥିର କର ? ଏଠାଶ୍ରଯ S = { 1,2,3,4,5,6} => ISI= 6 E1= ମୁଗମ ସଂଖ୍ୟା = { 2,4,6} => IE1I= 3 E2 ≥ 𝟒 = { 4,5,6} => IE2I= 3 E1 ∩ E2 = { 2,4,6} ∩ { 4,5,6} = {4,6} =>I E1 ∩ E2 I = 2 ପ ମୁଗମ ସଂଖ୍ୟା ଶ୍ରହଫା କିଭବା ≥ 𝟒 ଶ୍ରହଫା ଘଟଣା P(E1 ∪ E2 ) ଦ୍ୱାଯା ନିର୍ଣ୍ଥୟ ଶ୍ରହଫ I P ( E1 ∪ E2 ) = P (E1) + P (E2) - P(E1 ∩ E2) => P (E1 ∪ E2 ) = IE1I 𝑰𝑺𝑰 + IE𝟐I 𝑰𝑺𝑰 - I E1 ∩ E2I 𝑰𝑺𝑰 = 𝟑 𝟔 + 𝟑 𝟔 - 𝟐 𝟔 = 𝟒 𝟔 = 𝟐 𝟑 ରକ୍ଷକଯ ଏଠାଶ୍ରଯ IE1 ∪ E2 I = 4 P IE1 ∪ E2 I = - I E1∪E2I 𝑰𝑺𝑰 = = 𝟒 𝟔 = 𝟐 𝟑
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