Answer to Question #210133 in Electrical Engineering for Bahawal Tahir

Question #210133

Q2

if two events, A and B, are such that P(A) = .5, P(B) = .3, and P(A intersection B) = .1, findthe following:

a).P(A|B)

b). P(B|A)

c).P(A|AUB)

d).P(A|A intersection B)

e). P(A intersection B|A U B)


1
Expert's answer
2021-06-28T08:53:03-0400

a).

"P(A|B)=\\frac{P(A\\bigcap B)}{P(B)}\\\\\nP(A\\bigcap B)=0.1\\\\\nP(B)=0.3\\\\\nP(A|B)=\\frac{0.1}{0.3}=\\frac{1}{3}\\\\"

b).

"P(B|A)=\\frac{P(B\\bigcap A)}{P(A)}\\\\\nP(B\\bigcap A)=P(A\\bigcap B)=0.1\\\\\nP(A)=0.5\\\\\nP(B|A)=\\frac{0.1}{0.5}=\\frac{1}{5}\\\\"

c).

"P(A|A\\bigcup B)=\\frac{P(A\\bigcap (A\\bigcup B))}{P(A\\bigcup B)}\\\\\nP(A|A\\bigcup B)=\\frac{P(A)}{P(A\\bigcup B)}\\\\\nP(A)=0.5\\\\\nP(A\\bigcup B)=P(A)+P(B)-P(A\\bigcap B)\\\\\nP(A\\bigcup B)=0.5+0.3-0.1=0.7\\\\\nP(A|A\\bigcup B)=\\frac{0.5}{0.7}=\\frac{5}{7}\\\\"

d).

"P(A|A\\bigcap B)=\\frac{P(A\\bigcap (A\\bigcap B))}{P(A\\bigcap B)}\\\\\nP(A|A\\bigcap B)=\\frac{P(A\\bigcap B)}{P(A\\bigcap B)}=1\\\\"

e).

"P(A\\bigcap B|A \\bigcup B)=\\frac{P(A\\bigcap B\\bigcap A \\bigcup B)}{P(A \\bigcup B)}\\\\\nP(A\\bigcap B|A \\bigcup B)=\\frac{P(A\\bigcap B)}{P(A \\bigcup B)}\\\\\nP(A\\bigcap B|A \\bigcup B)=\\frac{0.1}{0.7}=\\frac{1}{7}\\\\"


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