Click here: Chapter Probability Part:1|Class 12|UGC CBCS| Statistics
Q11.Show that if a fair die is thrown twice getting 'five' in the first throw and getting 'six in the second throw are statistically independent.
Ans:
$\small A:$ getting five in the first throw. $\small P(A)=\frac{1}{6}$
$\small B:$ getting six in the second throw.$\small P(B)=\frac{1}{6}$
When a fair die throw twice the all possible out comes are 36.
(1,1),(1,2),......,(1,6)
(2,1),(2,2),.......,(2,6)
.
.
(5,1),(5,2),..(5,5),(5,6)
(6,1),(6,2),........,(6,6)
$\small P(A\cap B)=\frac{1}{36}=\frac{1}{6}.\frac{1}{6}=P(A)P(B)$
Q12.Each coefficient in the quadratic equation $\small ax^2+bx+c=0$ is determined by selecting randomly integers from 1,2,3,4,5. Find the probability that the equation will have
i.equal roots
ii. real roots
Ans: $\small ax^2+bx+c=0$ will have real roots if $\small b^2=4ac$
equal roots if$\small b^2\geq 4ac$
$\small a,b,c$ can take 1,2,3,4,5 these five values
So all possible outcomes are $\small 5^3=125$
Note: As we know the highest value of b=5, thats why $\small b^2_{max}=25$
Total: 24 4
$\small P(real-roots)$=$\frac{24}{125}$
$\small P(equal-roots)$=$\frac{4}{125}$
Q13.Suppose the chances of simultaneous occurence of two independent events A and B is $\small \frac{1}{6}$ and the probability that none of these two events occurs in $\small \frac{1}{3}$.Find $\small P(B)=?$
Let, P(A)=x
P(B)=y
$\small P(A\cap B)=xy=\frac{1}{6}$
=$\small xy=\frac{1}{6}$
$\small P(A^c \cap B^c)=\frac{1}{3}$
=$\small (1-x)(1-y)=\frac{1}{3}$
Now put $\small x=\frac{1}{6y}$
$\small y=\frac{1}{3},\frac{1}{2}$
Click here: Chapter Probability Part:3|Class 12|UGC CBCS| Statistics
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