I'm really struggling with this concept, hoping you guys could help me out.
Question: You have been provided with 20 pairs of socks within a box consisting of 4 red pairs, 4 yellow pairs, 4 green pairs, 4 blue pairs and 4 purple plairs.
The pairs have been separated out and you must take out a pair of socks.
Consider these problems and provide a calculation for each:
- Probability of drawing a matching pair if you randomly draw 2 socks?
- Probability of drawing a matching pair if you randomly draw 3 socks?
- (Repeats up to randomly drawing 5 socks)
For 2 socks I got the following:
40 possible socks * 39 other possible socks = 1560 possible combinations of socks / 2 (to remove duplicate matches) = 780
For each set of socks, there are 8. 8 * 7 (7 other socks to each being matched) = 56 possible combinations in each set of socks / 2 to remove duplicates = 28 possible combinations of socks in each set.
28 / 780 = 0.036 probability of drawing a pair when drawing 2 socks from the drawer.
I'm completely lost when it comes to drawing three socks from the drawer, however –
Cheers guys!
Best Answer
To pull two matching socks in two draws.
$\frac{5{4\choose2}}{20\choose2} = \frac{5*4*3}{20*19} = \frac{3}{19}$
To pull two matching socks in three draws.
you can pull 3 socks of the same color, or 2 socks of one color, and sock of annother color.
$\frac{5{4\choose2}*16 + 5*{4\choose3} }{20\choose3}$
we can keep going with this methodolgy to work up to 4 and 5
4 draws. $\frac{5{4\choose2}*16*12 + 5*{4\choose3}*16 + 5*{4\choose4} }{20\choose4}$
5 draws. $\frac{5{4\choose2}*16*12*8 + 5*{4\choose3}*16*12 + 5*{4\choose4}*16 }{20\choose5}$
However at 5 draws, it is easier to thing of the probablility of not getting a match.
$1 * \frac{16}{19} *\frac{12}{18}*\frac{8}{17}*\frac{4}{16}$