Answer:
The answer is "
".
Step-by-step explanation:
You have 4/10 opportunities to choose a white ball, and there'll be 7 white balls and 6 black balls out of 13, and so the second time they choose a white one is 7/13, as well as the second time they choose a black, 6/13. people will also have a 4/10 chance.
There are 6/10 chances which users pick its black ball and 4 white balls would still be picked, but 9 black balls and out 13 balls and thus, its second and third time you select the white one is 4/13 but you are likely to pick a black for the second time is 9/13.
Taking the diagram of the next tree. The very first draw is marked with a and the second draw is marked with b.
![\to P(a) = \frac{4}{10}\ \ \ \ \ \ \ \ \ P(b) = \frac{6}{10}\\\\\to P(\frac{a2}{a1}) = \frac{7}{13} \ \ \ \ \ \ \ \ \ \ P(\frac{a}{b}) = \frac{4}{13}\\\\\to P(\frac{b2}{a1}) = \frac{6}{13} \ \ \ \ \ \ \ \ \ \ P(\frac{b2}{b1}) = \frac{9}{13}](https://tex.z-dn.net/?f=%5Cto%20P%28a%29%20%3D%20%5Cfrac%7B4%7D%7B10%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20P%28b%29%20%3D%20%5Cfrac%7B6%7D%7B10%7D%5C%5C%5C%5C%5Cto%20P%28%5Cfrac%7Ba2%7D%7Ba1%7D%29%20%3D%20%5Cfrac%7B7%7D%7B13%7D%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20P%28%5Cfrac%7Ba%7D%7Bb%7D%29%20%3D%20%5Cfrac%7B4%7D%7B13%7D%5C%5C%5C%5C%5Cto%20P%28%5Cfrac%7Bb2%7D%7Ba1%7D%29%20%3D%20%5Cfrac%7B6%7D%7B13%7D%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20P%28%5Cfrac%7Bb2%7D%7Bb1%7D%29%20%3D%20%5Cfrac%7B9%7D%7B13%7D)
Calculating the second drawn ball is white:
![\to P(b2)=P(a)P(\frac{a2}{b1})+P(b)P(\frac{a}{b})\\](https://tex.z-dn.net/?f=%5Cto%20P%28b2%29%3DP%28a%29P%28%5Cfrac%7Ba2%7D%7Bb1%7D%29%2BP%28b%29P%28%5Cfrac%7Ba%7D%7Bb%7D%29%5C%5C)
![=\frac{4}{10}\frac{7}{13}+\frac{6}{10}\frac{4}{13}\\\\=\frac{28}{130}+\frac{24}{130}\\\\=\frac{28+24}{130}\\\\=\frac{52}{130}\\\\=\frac{2}{5}\\\\](https://tex.z-dn.net/?f=%3D%5Cfrac%7B4%7D%7B10%7D%5Cfrac%7B7%7D%7B13%7D%2B%5Cfrac%7B6%7D%7B10%7D%5Cfrac%7B4%7D%7B13%7D%5C%5C%5C%5C%3D%5Cfrac%7B28%7D%7B130%7D%2B%5Cfrac%7B24%7D%7B130%7D%5C%5C%5C%5C%3D%5Cfrac%7B28%2B24%7D%7B130%7D%5C%5C%5C%5C%3D%5Cfrac%7B52%7D%7B130%7D%5C%5C%5C%5C%3D%5Cfrac%7B2%7D%7B5%7D%5C%5C%5C%5C)
In point b:
![\to P(\frac{b}{a1})= \frac{P(B)P(\frac{a}{b})}{P(a)P(\frac{a2}{b1})+P(b)P(\frac{a}{b})\\}](https://tex.z-dn.net/?f=%5Cto%20P%28%5Cfrac%7Bb%7D%7Ba1%7D%29%3D%20%5Cfrac%7BP%28B%29P%28%5Cfrac%7Ba%7D%7Bb%7D%29%7D%7BP%28a%29P%28%5Cfrac%7Ba2%7D%7Bb1%7D%29%2BP%28b%29P%28%5Cfrac%7Ba%7D%7Bb%7D%29%5C%5C%7D)
![=\frac{\frac{6}{10} \frac{4}{13}}{\frac{52}{130}}\\\\=\frac{\frac{24}{130}}{\frac{52}{130}}\\\\=\frac{24}{130} \times \frac{130}{52}\\\\=\frac{24}{52}\\\\=\frac{6}{13}\\](https://tex.z-dn.net/?f=%3D%5Cfrac%7B%5Cfrac%7B6%7D%7B10%7D%20%5Cfrac%7B4%7D%7B13%7D%7D%7B%5Cfrac%7B52%7D%7B130%7D%7D%5C%5C%5C%5C%3D%5Cfrac%7B%5Cfrac%7B24%7D%7B130%7D%7D%7B%5Cfrac%7B52%7D%7B130%7D%7D%5C%5C%5C%5C%3D%5Cfrac%7B24%7D%7B130%7D%20%5Ctimes%20%5Cfrac%7B130%7D%7B52%7D%5C%5C%5C%5C%3D%5Cfrac%7B24%7D%7B52%7D%5C%5C%5C%5C%3D%5Cfrac%7B6%7D%7B13%7D%5C%5C)