Question
Evaluate $\text{P}(\text{A}\cap\text{B})\ \text{if}\ 2\text{P}(\text{A})=\text{P}(\text{B})=\frac{5}{13}\ \text{and}\ \text{P}(\text{A}|\text{B})=\frac{2}{5}.$
✓
Answer
$\text{Given:}\ \ \ \ \ 2\text{P}\left(\text {A}\right)=\text{P}\left(\text{B}\right)=\frac{5}{13},\ \text{P}\left(\text{A}|\text{B}\right)=\frac{2}{5}$
$ \therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \text{P}\left(\text{A}\right)=\frac{5}{26}$
$\text{Now}\ \ \ \ \ \ \ \ \ \text{P}\left (\text{A}\cap\text{B}\right)=\text{P}\left(\text{B}|\text{A}\right).\text{P}\left(\text{B}\right)=\frac{2}{5}\times\frac{5}{13}=\frac {2}{13}$
$\text{And}\ \ \ \ \ \ \ \ \ \ \text{P}\left (\text{A}\cup\text{B}\right)=\text{P}\left(\text{A}\right)+\text{P}\left(\text{B}\right)-\text{P}\left(\text{A}\cap\text{B}\right)= \frac{5}{26}+\frac{5}{13}-\frac{2}{13}=\frac{11}{26}$
$ \therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \text{P}\left(\text{A}\right)=\frac{5}{26}$
$\text{Now}\ \ \ \ \ \ \ \ \ \text{P}\left (\text{A}\cap\text{B}\right)=\text{P}\left(\text{B}|\text{A}\right).\text{P}\left(\text{B}\right)=\frac{2}{5}\times\frac{5}{13}=\frac {2}{13}$
$\text{And}\ \ \ \ \ \ \ \ \ \ \text{P}\left (\text{A}\cup\text{B}\right)=\text{P}\left(\text{A}\right)+\text{P}\left(\text{B}\right)-\text{P}\left(\text{A}\cap\text{B}\right)= \frac{5}{26}+\frac{5}{13}-\frac{2}{13}=\frac{11}{26}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the distance between the planes 2x – y + 2z = 5 and 5x – 2.5y + 5z = 20.
In the figure which are the Equal vectors?
Find $\int x \cos x d x$
→Find the area bounded by lines $y=x, x=4$ and $x$-axis.
→State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
| Z | 3 | 2 | 1 | 0 | -1 |
| P(Z) | 0.3 | 0.2 | 0.4 | 0.1 | 0.05 |
If $\triangle = \begin{vmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{vmatrix}$, write the minor of the element a23.
→Write the degree of the differrntial equation $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+\text{e}^{\frac{\text{dy}}{\text{dx}}}=0.$
→Find the distance between the planes 2x – y + 2z = 5 and 5x – 2.5y + 5z = 20.
Evaluate:
→$\int\limits_0^1\frac{\text{2x}}{\text{1 + x}^{2}}\text{dx}$.
Show that the function given by f (x) = sin x is increasing in $\left(0, \frac{\pi}{2}\right)$
→