Question

a bin contains 6 blue socks and 4 black socks. you randomly select 3 socks from the bin. you do not replace each sock before selecting the next one. what is the probability that all three socks are black for each situation?
( \frac{2}{5} \times \frac{1}{3} \times \frac{1}{4} )
( \frac{4}{10} \times \frac{4}{10} \times \frac{4}{10} )
( \frac{4 \times 3 \times 2}{100} )
( \frac{4}{10} \times \frac{6}{10} \times \frac{4}{6} )

Explanation:

Step1: Analyze total and black socks

Total socks: \(6 + 4 = 10\). First black sock probability: \(\frac{4}{10}=\frac{2}{5}\).

Step2: After first black, remaining socks

Now 9 socks, 3 black. Second black probability: \(\frac{3}{9}=\frac{1}{3}\).

Step3: After second black, remaining socks

Now 8 socks, 2 black. Third black probability: \(\frac{2}{8}=\frac{1}{4}\).

Step4: Multiply probabilities

Probability all three black: \(\frac{2}{5} \times \frac{1}{3} \times \frac{1}{4}\).

Answer:

\(\boldsymbol{\frac{2}{5} \times \frac{1}{3} \times \frac{1}{4}}\) (the first option)