Question

fred reaches into a bag of 40 jumbled pairs of socks, in which there are 20 black socks, 16 brown ones and 4 blue ones. if he takes two socks without replacement, what is the probability that they are black?

Explanation:

Step1: Calculate probability of first black sock

There are 20 black socks out of 40 total. So, \( P(\text{1st black}) = \frac{20}{40} \).

Step2: Calculate probability of second black sock (without replacement)

After taking one black sock, there are 19 black socks left and 39 total socks. So, \( P(\text{2nd black} | \text{1st black}) = \frac{19}{39} \).

Step3: Multiply the two probabilities

To find the probability of both events (first and second sock black), multiply the two probabilities: \( P = \frac{20}{40} \times \frac{19}{39} \).
Simplify \( \frac{20}{40} = \frac{1}{2} \), then \( \frac{1}{2} \times \frac{19}{39} = \frac{19}{78} \).

Answer:

\(\frac{19}{78}\)