Answer:
The probability that A selects the first red ball is 0.5833.
Step-by-step explanation:
Given : An urn contains 3 red and 7 black balls. Players A and B take turns (A goes first) withdrawing balls from the urn consecutively.
To find : What is the probability that A selects the first red ball?
Solution :
A wins if the first red ball is drawn 1st,3rd,5th or 7th.
A red ball drawn first, there are
places in which the other 2 red balls can be placed.
A red ball drawn third, there are
places in which the other 2 red balls can be placed.
A red ball drawn fifth, there are
places in which the other 2 red balls can be placed.
A red ball drawn seventh, there are
places in which the other 2 red balls can be placed.
The total number of total event is
The probability that A selects the first red ball is
![P(A \text{wins})=\frac{(^9C_2)+(^7C_2)+(^5C_2)+(^3C_2)}{^{10}C_3}](https://tex.z-dn.net/?f=P%28A%20%5Ctext%7Bwins%7D%29%3D%5Cfrac%7B%28%5E9C_2%29%2B%28%5E7C_2%29%2B%28%5E5C_2%29%2B%28%5E3C_2%29%7D%7B%5E%7B10%7DC_3%7D)
![P(A \text{wins})=\frac{36+21+10+3}{120}](https://tex.z-dn.net/?f=P%28A%20%5Ctext%7Bwins%7D%29%3D%5Cfrac%7B36%2B21%2B10%2B3%7D%7B120%7D)
![P(A \text{wins})=\frac{70}{120}](https://tex.z-dn.net/?f=P%28A%20%5Ctext%7Bwins%7D%29%3D%5Cfrac%7B70%7D%7B120%7D)
![P(A \text{wins})=0.5833](https://tex.z-dn.net/?f=P%28A%20%5Ctext%7Bwins%7D%29%3D0.5833)