To find the probability that Emily selects two differently colored socks from a bag containing four tan socks and seven gray socks, we first need to calculate the total number of ways to select two socks, then the number of ways to select one tan sock and one gray sock.
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Calculate the total number of socks: \[ \text{Total socks} = 4 \text{ (tan)} + 7 \text{ (gray)} = 11 \text{ socks} \]
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Calculate the total ways to choose 2 socks from 11: The number of ways to choose 2 socks from 11 is given by the combination formula \( \binom{n}{k} \), which is calculated as: \[ \binom{11}{2} = \frac{11 \times 10}{2 \times 1} = 55 \]
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Calculate the number of ways to choose one tan sock and one gray sock:
- The number of ways to choose 1 tan sock from 4 is \( \binom{4}{1} = 4 \).
- The number of ways to choose 1 gray sock from 7 is \( \binom{7}{1} = 7 \).
Therefore, the total ways to choose one tan and one gray sock is: \[ \text{Ways to choose 1 tan and 1 gray} = \binom{4}{1} \times \binom{7}{1} = 4 \times 7 = 28 \]
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Calculate the probability of choosing differently colored socks: The probability \( P(\text{different colors}) \) is given by the ratio of the number of successful outcomes (choosing one tan and one gray sock) to the total number of outcomes (choosing any two socks): \[ P(\text{different colors}) = \frac{\text{Number of ways to choose differently colored socks}}{\text{Total ways to choose 2 socks}} = \frac{28}{55} \]
Therefore, the probability that Emily will get two differently colored socks is: \[ \boxed{\frac{28}{55}} \]
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