Ang Imhr.ca ay narito upang tulungan kang makahanap ng mga sagot sa lahat ng iyong mga katanungan mula sa mga eksperto. Kumonekta sa isang komunidad ng mga eksperto na handang tumulong sa iyo na makahanap ng eksaktong solusyon sa iyong mga tanong nang mabilis at mahusay. Maranasan ang kadalian ng paghahanap ng eksaktong sagot sa iyong mga tanong mula sa isang malawak na komunidad ng mga eksperto.
Sagot :
[tex] \large \bold\colorbox{gray}{PROBLEM:} [/tex]
Suppose there are two urns, A and B, containing white and yellow balls. Urn A contains 3 white balls and 2 yellow balls, while Urn B contains 2 white balls and 3 yellow balls.
[tex] \bold\colorbox{gray}{Q1.} [/tex] A random ball is transferred from Urn A to Urn B without knowing its color. If a ball is chosen randomly from Urn B after the transfer, what is the probability that it will be a white ball?
[tex] \bold\colorbox{gray}{SOLUTION:} [/tex]
[tex] \begin{array}{l} \textsf{Let the event that a white ball is drawn from urn} \\ B\textsf{ denoted as }W_B. \\ \\ \textsf{Now we have to consider 2 cases.} \\ \\ \bold{Case\ 1:}\textsf{ A white ball is drawn from urn }A. \\ \\ \textsf{The probability of drawing a white ball from urn} \\ A\textsf{ and then a white ball from urn }B: \\ \\ \quad \quad \dfrac{3}{3 + 2} \times \dfrac{2 + 1}{2 + 3 + 1} = \dfrac{3}{10} \\ \\ \\ \bold{Case\ 2:}\textsf{ A yellow ball is drawn from urn }A. \\ \\ \textsf{The probability of drawing a yellow ball from} \\ \textsf{urn }A\textsf{ and then a white ball from urn }B: \\ \\ \quad \quad \dfrac{2}{3 + 2} \times \dfrac{2}{2 + 3 + 1} = \dfrac{2}{15} \\ \\ \textsf{Thus, the overall probability is}: \\ \\ \quad \: \: P(W_B) = \dfrac{3}{10} + \dfrac{2}{15} = \boxed{\dfrac{13}{30}}\:\:\:\textit{Answer} \end{array} [/tex]
[tex] \bold\colorbox{gray}{Q2.}[/tex] Suppose 2 balls are transferred from Urn A to Urn B without knowing its color. If a ball is chosen randomly from Urn B after the transfer, what is the probability that it will be a white ball?
[tex] \bold\colorbox{gray}{SOLUTION:} [/tex]
[tex] \begin{array}{l} \textsf{Let the event that a white ball is drawn from urn} \\ B\textsf{ denoted as }W_B. \\ \\ \textsf{Now we have to consider 3 cases.} \\ \\ \bold{Case\ 1:}\textsf{ 2 white balls are drawn from urn }A. \\ \\ \textsf{The probability of drawing 2 white balls from} \\ \textsf{urn }A\textsf{ and then a white ball from urn }B: \\ \\ \quad \dfrac{3}{3 + 2} \times \dfrac{3 - 1}{3 + 2 - 1} \times \dfrac{2 + 2}{2 + 3 + 2} = \dfrac{6}{35} \\ \\ \\ \bold{Case\ 2:}\textsf{ A white ball and a yellow ball are drawn} \\ \qquad \quad\:\:\: \textsf{from urn }A. \\ \\ \textsf{The probability of drawing a white ball and a} \\ \textsf{yellow ball from urn }A\textsf{ and then a white ball} \\ \textsf{from urn }B: \\ \\ \quad \dfrac{3}{3 + 2} \times \dfrac{2}{3 + 2 - 1} \times \dfrac{2 + 1}{2 + 3 + 2} = \dfrac{9}{70} \\ \\ \bold{Case\ 3:}\textsf{ 2 yellow balls are drawn from urn }A. \\ \\ \textsf{The probability of drawing 2 yellow balls from} \\ \textsf{urn }A\textsf{ and then a white ball from urn }B: \\ \\ \quad \dfrac{2}{3 + 2} \times \dfrac{2 - 1}{3 + 2 - 1} \times \dfrac{2}{2 + 3 + 2} = \dfrac{1}{35} \\ \\ \textsf{Thus, the overall probability is}: \\ \\ \:\:\: P(W_B) = \dfrac{6}{35} + \dfrac{9}{70} + \dfrac{1}{35} = \boxed{\dfrac{23}{70}}\:\:\:\textit{Answer} \end{array} [/tex]
[tex] \mathfrak \colorbox{gray}{\#CarryOnLearning} [/tex]
Salamat sa pagtitiwala sa amin sa iyong mga katanungan. Narito kami upang tulungan kang makahanap ng tumpak na mga sagot nang mabilis at mahusay. Salamat sa pagpili sa aming plataporma. Kami ay nakatuon sa pagbibigay ng pinakamahusay na mga sagot para sa lahat ng iyong mga katanungan. Bisitahin muli kami. Maraming salamat sa paggamit ng Imhr.ca. Bumalik muli para sa karagdagang kaalaman mula sa aming mga eksperto.
