Math proof required (Probability) -

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from pr[e] = pr[e|a].pr[a] + pr[e|a'].pr[a']

how can prove pr[e] <= pr[e|a] + pr[a']

pr[e] <= pr[e|a] + pr[a']

the left side can replaced first line..

pr[e|a].pr[a] + pr[e|a'].pr[a'] <= pr[e|a] + pr[a']

hmm lets subtract "pr[e|a].pr[a]" on both sides. on right side, can translate pr[e|a] = pr[e|a]*1 = pr[e|a] (pr[a] + pr[a'])

pr[e|a'].pr[a'] <= pr[e|a].pr[a'] + pr[a']

now can put both sides brackets in order isolate pr[a']

( pr[e|a'] ) * pr[a'] <= ( pr[e|a] + 1 ) * pr[a']

and divide pr[a']

pr[e|a'] <= pr[e|a] + 1

so.. if.. pr[e|a] = 0 both sides can equal (if left side 1) in other cases right side larger, since more 1, , left side can maximum 1


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