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Brent starts with \$16\$ identical white socks in his sock drawer. Imagine he receives \$2\$ identical black socks as a gift and mixes them in with his \$16\$ white socks. If he draws one sock without looking to put on his left foot, then draws a second sock without looking to put on his right foot, what is the probability that he draws mismatched socks?

Jul 26, 2023

#1
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The probability of Brent drawing mismatched socks is the sum of the probabilities of two mutually exclusive events: drawing a white sock on the left foot and a black sock on the right foot, or drawing a black sock on the left foot and a white sock on the right foot.

The probability of drawing a white sock on the left foot is 18/20​, since there are 18 total socks and 18 white socks.

The probability of drawing a black sock on the right foot is 2/18​, since there are 2 black socks and 18 total socks remaining after the first sock is drawn.

Therefore, the probability of drawing a white sock on the left foot and a black sock on the right foot is 18/20​*2/18​=1/5​.

Similarly, the probability of drawing a black sock on the left foot and a white sock on the right foot is also 1/5.

Therefore, the probability of Brent drawing mismatched socks is 1/5 + 1/5 = 2/5.

Hope this helps!

Jul 26, 2023
#2
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Umm.... I think that's wrong because there are WAY less black socks than white, so I think the probability should be smaller, but thanks anyway.

Jul 26, 2023
edited by Guest  Jul 26, 2023
#3
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Hmmm. So, we can see that there are 16 white socks and 2 black socks. There are 18 socks in total. 16 / 18 that he draws white. Then, we multiply 16 / 18 by 2 / 17. It is 2 / 17 instead of 2 / 18 because we already taken a white sock. When we multiply them together, we get 16 / 153. Then, we do it the other way around. We have a 2 / 18 chance of getting a black sock. Then, we have a 16 / 17 chance of getting a white sock. When we multiply them together, we get 16 / 153 again. Therefore, we multiply 16 / 153 * 2, or 32 / 153

Jul 26, 2023
#4
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If Brent draws a white sock with his left foot, then he can draw either a white or a black sock with his right foot to have mismatched socks. Likewise, if he draws a black sock with his left foot, he can draw a white or a black sock with his right foot to have mismatched socks. So, there are 2×2=4 ways for Brent to draw mismatched socks.

The total number of possible outcomes is 18×18=324 because there are 18 socks total and 18 possibilities for each draw. So, the probability that Brent draws mismatched socks is 4/324 = 1/81,

Jul 27, 2023
#5
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I'm not sure how you got that answer. First off, you can not get two black socks. Second, there should be a much higher chance the 1 / 81

history  Jul 27, 2023
#6
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There are 18 total socks in the drawer, and 2 of them are black. There are two ways to draw mismatched socks: he can draw a black sock on the left foot and a white sock on the right foot, or he can draw a white sock on the left foot and a black sock on the right foot.

The probability of drawing a black sock on the left foot and a white sock on the right foot is 2/18​⋅16/17​=8/153​.

The probability of drawing a white sock on the left foot and a black sock on the right foot is also 8/153.

So, the probability of drawing mismatched socks is 8/153​+8/1538​=16/153​​.

Jul 28, 2023