Answer:
The probability that the maximum number of draws is required is 0.2286
Step-by-step explanation:
The probability that the maximum number of draws happens when you pick different colors in the first four pick.
Assume you picked one sock in the first draw. Its probability is 1, since you can draw any sock.
In the second draw, 7 socks left and you can draw all but the one which is the pair of the first draw. Then the probability is [tex]\frac{6}{7}[/tex]
In the third draw, 6 socks left and you can draw one of the two pair colors which are not drawn yet. Its probability is [tex]\frac{4}{6}[/tex]
In the forth draw, 5 socks left and only one pair color, which is not drawn. The probability of drawing one of this pair is [tex]\frac{2}{5}[/tex]
In the fifth draw, whatever you draw, you would have one matching pair.
The probability combined is 1×[tex]\frac{6}{7}[/tex] ×[tex]\frac{4}{6}[/tex]× [tex]\frac{2}{5}[/tex] ≈ 0.2286
