91影视

There are 3 red and 5 black balls in one box and 6 red and 4 white balls in another box.

The events are said to be independent when the occurrence or non-occurrence of any event does not have any effect on the occurrence or non-occurrence of the other event.

There are total 8balls in first box out of which 3 are red and total 10red balls are in the second out of which 6 are red and each box has a probability $\frac{1}{2}$of getting picked.

Find the probability of picking a red ball from any box.

$$\begin{gathered} P\left(Red\right)=\frac{1}{2}脳\frac{3}{8}I\frac{1}{2}脳\frac{6}{10} \\ =\frac{39}{80} \end{gathered}$$

Find the probability of picking a black ball from any box.

$$\begin{gathered} P\left(Black\right)=\frac{1}{2}+\frac{5}{8}脳\frac{0}{10} \\ =\frac{5}{16} \end{gathered}$$

Find the probability of picking a white ball from any box.

$$\begin{gathered} P\left(White\right)=\frac{1}{2}脳\frac{0}{8}I\frac{1}{2}脳\frac{4}{10} \\ =\frac{1}{5} \end{gathered}$$

Find the probability of pickingeither red or white ball from any box.

$$\begin{gathered} P\left(Red+White\right)=\frac{1}{2}脳\frac{3}{8}+\frac{1}{2}脳\left(\frac{6}{10}+\frac{4}{10}\right) \\ =\frac{3}{16}+\frac{1}{2} \\ =\frac{11}{16} \end{gathered}$$

Find The probability that the second ball is also red when first ball is redusing the Bayes theorem, $P_A\left(B\right)=\frac{P\left(AB\right)}{P\left(A\right)}$.

$$\begin{gathered} P\left(BothRedIOneRed\right)=\frac{{\displaystyle \frac{1}{2}}脳{\displaystyle \frac{3}{8}}脳{\displaystyle \frac{1}{2}}+脳{\displaystyle \frac{6}{10}}脳{\displaystyle \frac{5}{9}}}{{\displaystyle \frac{39}{80}}} \\ =\frac{370}{819} \end{gathered}$$

Thus, the desired probability is $\frac{370}{819}$.

Find the probability of getting both reds.

$\mathit{P}\left(BothReds\right)\mathbf{=}\frac{\mathbf{3}}{\mathbf{8}}\mathbf{脳}\frac{\mathbf{2}}{\mathbf{7}}\mathbf{+}\frac{\mathbf{3}}{\mathbf{8}}\mathbf{脳}\frac{\mathbf{6}}{\mathbf{10}}\mathbf{+}\frac{\mathbf{6}}{\mathbf{10}}\mathbf{脳}\frac{\mathbf{3}}{\mathbf{10}}\mathbf{+}\frac{\mathbf{6}}{\mathbf{10}}\mathbf{脳}\frac{\mathbf{5}}{\mathbf{9}}$

Find the probability of getting both reds from same box.

$\mathit{P}\left(BothRedsfromsamebox\right)\mathbf{=}\frac{\mathbf{3}}{\mathbf{8}}\mathbf{脳}\frac{\mathbf{2}}{\mathbf{7}}\mathbf{+}\frac{\mathbf{6}}{\mathbf{10}}\mathbf{脳}\frac{\mathbf{5}}{\mathbf{9}}$

Find the probability that they both came from the same box, when both balls are red

Using the Bayes theorem, $P_A\left(B\right)=\frac{P\left(AB\right)}{P\left(A\right)}$.

role="math" localid="1654839283298" $$\begin{gathered} \mathit{P}\left(BothRedsfromsameboxIBothRed\right)\mathbf{=}\frac{{\displaystyle \frac{3}{8}脳\frac{2}{7}+\frac{6}{10}脳\frac{5}{9}}}{{\displaystyle \frac{3}{8}脳\frac{2}{7}+\frac{3}{8}脳\frac{6}{10}+\frac{6}{10}脳\frac{3}{8}+\frac{6}{10}脳\frac{5}{9}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{鈮�}\frac{\mathbf{1}}{\mathbf{2}} \end{gathered}$$