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Important Examples
Q.If P(A)=a and P(B)=b, show that \small P(A|B)\leq \frac{a}{b}
Ans: \small P(A\cap B) \leq P(A)
So, \small P(A/B)=\frac {P(A \cap B)}{P(B)}
\small \frac {P(A \cap B)}{P(B)} \leq \frac {P(A)}{P(B)} \leq \frac{a}{b}
Q. If \small P(A|B)=1 then prove that \small P(ABC)=P(BC)
Ans: \small P(A/B)=\frac{P(A\cap B)}{P(B)}=1
\small P(A\cap B)=P(B)
\small P(ABC)=P(A \cap B \cap C)= P(B \cap C) [Since \small P(A\cap B)=P(B)]
Q.Lots 1,2 and 3 contain equal number of manufactured items among which 30%,24% and 20% respectively are defective. In throwing an unbiased coin once, one item is selected from lot number 1 if 1 comes, one item is selected from lot number 2 if 2 or 3 comes, and one item is selected from lot number 3 otherwise. Find the probability that the selected item is defected.
Ans: Let, \small A_i is the event that the person will select \small i'th number of box
\small D is the event of selecting the defective item
\small P(D)=\sum P(A_i \cap D)
\small P(A_1)=\frac{1}{6}
\small P(A_2)=\frac{2}{6}
\small P(A_3)=\frac{3}{6}
We know \small P(A_i \cap D)=\sum P(D/A_i)P(A_i)
\small P(D/A_1)=0.30
\small P(D/A_1)=0.24
\small P(D/A_1)=0.20
Now calculate: \small P(A_i \cap D)=\sum P(D/A_i)P(A_i)=0.23
Q.A speaks truth in 70% cases and his friend B speaks lie in 20% cases. In which percentage of cases they likely to contradict each other in narrating the same incident?
Ans: X denotes the event that A will lie \small P(X)=1-0.70=0.30,\small P(X^c)=0.70
Y denotes the event that B will lie \small P(Y)=0.20,\small P(Y^c)=1-0.20=0.80
\small P(X^c \cap Y)+P(X \cap Y^c)=\small (0.70 \times 0.20)+(0.80 \times 0.30)=0.38
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