how do you solve this problem? Your sock drawer contains 3 pairs of gray socks, 4 pairs of white socks, 4 pairs of black socks, and 7 pairs of beige socks. You choose a pair of socks from the drawer at random and then replace it. Then you choose a second pair of socks. Find P[gray,thengray].

STEP 1:
First, calculate P[gray], which represents the probability of pulling out a pair of gray socks on the first try. To do this, you just need to calculate the percentage of gray pairs in the drawer and then use the formula:

(number pairs gray / number pairs total) * 100 [where / means divide and * means multiply]

STEP 1-A:
3 pairs gray + 4 pairs white + 4 pairs black + 7 pairs beige = 18 pairs in total

STEP 1-B:
3 pairs gray divided by 18 pairs total (or 3 / 18) = 0.166666667

STEP 1-C:
0.166666667 * 100 = 16.6666667% <-- this is P[gray] or the probability of pulling out a gray pair on the first try

STEP 2:
Next calculate P[gray, thengray], which represents the probability of pulling out a pair of gray socks on the first try and then again on the second try. To do this, you just need to multiply P[gray] by itself 2 times, because you want to calculate the probability of pulling out a gray pair 2 times in a row. Use this formula:

P[gray] * P[gray]
This can also be written as: "P[gray] squared" or "P[gray] to the second power"

STEP 2-A:
16.6666667% * 16.6666667% = 0.0277777779

STEP 2-B:
0.0277777779 * 100 = 2.77777779% <-- this is P[gray, thengray] or the probability of pulling out a gray pair on the first try and then again on the second try

TO SUMMARIZE:
P[gray] = 16.6666667% (the probability of pulling out a gray pair on the first try)
P[gray, thengray] = 2.77777779% (the probability of pulling out a gray pair on the first try and then again on the second try)