The number of purple pen is 2
Explanation:
Number of red pens = 3
Number of black pens = 3
Number of blue pens = 2
Number of purple pens = ?
Probability = [tex]\frac{1}{15}[/tex]
Total number of pens = 3 + 3 + 2 + x
= 8 + x
The probability of pulling out a red pen = [tex]\frac{3}{8+x}[/tex]
Total number of pens become = 8 + x - 1
= 7 + x
Probability of pulling out a purple pen = [tex]\frac{x}{7+x}[/tex]
According to the question:
[tex]\frac{3}{8+x} X\frac{x}{7+x} = \frac{1}{15}[/tex]
Solving the equation:
[tex]\frac{3x}{8(7+x) + x (7+x)} = \frac{1}{15} \\\\\frac{3x}{56 + 8x + 7x + x^2} = \frac{1}{15} \\[/tex]
[tex]\frac{3x}{56+ 15x + x^2} = \frac{1}{15} \\\\45x = 56 + 15x + x^2\\\\x^2 - 30x + 56 = 0\\\\x = 28, 2[/tex]
If x = 2 then,
[tex]\frac{3}{8 +2} X \frac{2}{7+2} = \frac{3}{10} X \frac{2}{9} = \frac{1}{15}[/tex]
Therefore, the number of purple pen is 2
