Click here: Probability Formulas
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 $\small P(Y)=0.20$,$\small P(Y^c)=1-0.20=0.80$
$\small P(X^c \cap Y)+P(X \cap Y^c)$=$\small (0.70 \times 0.20)+(0.80 \times 0.30)=0.38$
Click here to See: Probability Example(Part-2)|Class 12|UGC CBCS| Statistic
Tags: probability questions and answers pdf, probability problems on balls with solutions, real life probability problmes with solutions