Answer:
[tex]Probability = \frac{70}{143}[/tex]
Step-by-step explanation:
Given
[tex]Black = 5[/tex]
[tex]Red = 8[/tex]
[tex]Total = 13[/tex]
To solve the required probability, we need to determine the number of available selections:
There are two possible scenarios;
1: All 4 selections are red.
2. 1 selection is black, while 3 others are red
Scenario 1: All Red
[tex]^8C_4 = \frac{8!}{(8-4)!4!}[/tex]
[tex]^8C_4 = \frac{8!}{4!4!}[/tex]
[tex]^8C_4 = \frac{8*7*6*5*4!}{4!4!}[/tex]
[tex]^8C_4 = \frac{8*7*6*5}{4!}[/tex]
[tex]^8C_4 = \frac{8*7*6*5}{4*3*2*1}[/tex]
[tex]^8C_4 = \frac{1680}{24}[/tex]
[tex]^8C_4 = 70[/tex]
Scenario 2: 1 Black, 3 Red
Selecting Black:
[tex]^5C_1 = \frac{5!}{(5-1)!1!}[/tex]
[tex]^5C_1 = \frac{5!}{4!1!}[/tex]
[tex]^5C_1 = \frac{5*4!}{4!*1}[/tex]
[tex]^5C_1 = 5[/tex]
Selecting Red:
[tex]^8C_3 = \frac{8!}{(8-3)!3!}[/tex]
[tex]^8C_3 = \frac{8!}{5!3!}[/tex]
[tex]^8C_3 = \frac{8*7*6*5!}{5!3*2*1}[/tex]
[tex]^8C_3 = \frac{8*7*6}{6}[/tex]
[tex]^8C_3 = 8*7[/tex]
[tex]^8C_3 = 56[/tex]
Number of Selection = 5 * 56
[tex]Selection = 280[/tex]
Total Available Selection is calculated as thus:
[tex]Available\ Selection = 70 + 280[/tex]
[tex]Available\ Selection= 350[/tex]
Next, is to calculate the number of possible selections:
i.e 4 balls out of 13
This is calculated as:
[tex]^{13}C_4 = \frac{13!}{(13-4)!4!}[/tex]
[tex]^{13}C_4 = \frac{13!}{9!4!}[/tex]
[tex]^{13}C_4 = \frac{13 * 12 * 11 * 10 * 9!}{9!4*3*2*1}[/tex]
[tex]^{13}C_4 = \frac{13 * 12 * 11 * 10}{4*3*2*1}[/tex]
[tex]^{13}C_4 = \frac{17160}{24}[/tex]
[tex]^{13}C_4 = 715[/tex]
[tex]Probability = \frac{Available\ Selection}{Possible\ Selection}[/tex]
[tex]Probability = \frac{350}{715}[/tex]
[tex]Probability = \frac{70}{143}[/tex]
