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Probability

15 Probability

PROBABILITY

Experiment: Any activity or trial performed is an experiment.

e.g. 1) Tossing of coin

2) Rolling of dice

3) Falling of rain

 

·      Outcomes: The results of the activity are called outcomes.

e. g. out comes for above experiments are

1) Head, Tail

2) 1, 2, 3, 4, 5, 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeymaiaabYcacaqGGaGaaeOmaiaabYcacaqGGaGaae4maiaabYca caqGGaGaaeinaiaabYcacaqGGaGaaeynaiaabYcacaqGGaGaaeOnaa aa@41D5@

3) It will rain, It will not rain.

 

 

·      Probability of happening of event

P( E ) =  no.of trials in which event happened Total no.of trials MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabcfadaqada qaaiaabweaaiaawIcacaGLPaaacaqGGaGaaeypaiaabccadaWcaaqa aiaab6gacaqGVbGaaeOlaiaab+gacaqGMbGaaeiiaiaabshacaqGYb GaaeyAaiaabggacaqGSbGaae4CaiaabccacaqGPbGaaeOBaiaabcca caqG3bGaaeiAaiaabMgacaqGJbGaaeiAaiaabccacaqGLbGaaeODai aabwgacaqGUbGaaeiDaiaabccacaqGObGaaeyyaiaabchacaqGWbGa aeyzaiaab6gacaqGLbGaaeizaaqaaiaabsfacaqGVbGaaeiDaiaabg gacaqGSbGaaeiiaiaab6gacaqGVbGaaeOlaiaab+gacaqGMbGaaeii aiaabshacaqGYbGaaeyAaiaabggacaqGSbGaae4Caaaaaaa@6BC3@

If total no. of trials = n

If no. of times event E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabweaaaa@379C@ occurred = m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeypaiaabccacaqGTbaaaa@3947@

Then P( E ) =  m n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabcfadaqada qaaiaabweaaiaawIcacaGLPaaacaqGGaGaaeypaiaabccadaWcaaqa aiaab2gaaeaacaqGUbaaaaaa@3DEF@

 

·      Probability always lies between 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeimaaaa@37A7@ and v.

·      0p( E )1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabcdacqGHKj YOcaqGWbWaaeWaaeaacaqGfbaacaGLOaGaayzkaaGaeyizImQaaeym aaaa@3EE9@

·      If E 1 , E 2 , E 3 ... , E n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabweadaWgaa WcbaGaaeymaaqabaGccaqGSaGaaeiiaiaabweadaWgaaWcbaGaaeOm aaqabaGccaqGSaGaaeiiaiaabweadaWgaaWcbaGaae4maaqabaGcca qGSaGaaeiiaiaab6cacaqGUaGaaeOlaiaabYcacaqGGaGaaeyramaa BaaaleaacaqGUbaabeaaaaa@452D@ are the possible outcomes of an experiments, then sum of probabilities of each event is 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeymaaaa@37A8@ .

 

i.e P( E 1 ) + P( E 2 ) + P( E 3 ) +...+P( E n ) = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabcfadaqada qaaiaabweadaWgaaWcbaGaaeymaaqabaaakiaawIcacaGLPaaacaqG GaGaae4kaiaabccacaqGqbWaaeWaaeaacaqGfbWaaSbaaSqaaiaabk daaeqaaaGccaGLOaGaayzkaaGaaeiiaiaabUcacaqGGaGaaeiuamaa bmaabaGaaeyramaaBaaaleaacaqGZaaabeaaaOGaayjkaiaawMcaai aabccacaqGRaGaaeOlaiaab6cacaqGUaGaae4kaiaabcfadaqadaqa aiaabweadaWgaaWcbaGaaeOBaaqabaaakiaawIcacaGLPaaacaqGGa GaaeypaiaabccacaqGXaaaaa@5200@

 

e.g.

For an experiment rolling a dice

The possible out comes = Numbers - - 1, 2, 3, 4, 5, 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeylaiaabccacaqGXaGaaeilaiaabccacaqGYaGaaeilaiaabcca caqGZaGaaeilaiaabccacaqG0aGaaeilaiaabccacaqG1aGaaeilai aabccacaqG2aaaaa@4328@ So let event E 1 = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeyra8aadaWgaaWcbaWdbiaabgdaa8aabeaak8qacaqG9aaaaa@39A4@ getting 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeymaaaa@37A8@ on die, E 2 = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeyra8aadaWgaaWcbaGaaGOmaaqabaGcpeGaaeypaaaa@398D@ getting 2 on die and so on.

Then P( E 1 ) + P( E 2 ) + P( E 3 ) +...+P( E 6 ) = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabcfadaqada qaaiaabweadaWgaaWcbaGaaeymaaqabaaakiaawIcacaGLPaaacaqG GaGaae4kaiaabccacaqGqbWaaeWaaeaacaqGfbWaaSbaaSqaaiaabk daaeqaaaGccaGLOaGaayzkaaGaaeiiaiaabUcacaqGGaGaaeiuamaa bmaabaGaaeyramaaBaaaleaacaqGZaaabeaaaOGaayjkaiaawMcaai aabccacaqGRaGaaeOlaiaab6cacaqGUaGaae4kaiaabcfadaqadaqa aiaabweadaWgaaWcbaGaaeOnaaqabaaakiaawIcacaGLPaaacaqGGa GaaeypaiaabccacaqGXaaaaa@51C8@

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