In Sal's sock drawer, there are 2 red, 2 blue, 2 yellow, 2 black, and 2 green socks. If he randomly picks one sock from the drawer, and then another, which fractions should Sal multiply to find the probability of choosing a pair of green socks?
$\left(\frac{1}{10}\right)\left(\frac{1}{9}\right)$
$\left(\frac{2}{10}\right)\left(\frac{2}{10}\right)$
$\left(\frac{2}{10}\right)\left(\frac{1}{9}\right)$
$\left(\frac{2}{10}\right)\left(\frac{2}{9}\right)$
Answer (2)
The probability of picking a green sock on the first draw is 10 2 .
After picking one green sock, the probability of picking another green sock on the second draw is 9 1 .
Multiply these probabilities to find the overall probability: 10 2 × 9 1 .
The fractions to multiply are: ( 10 2 ) ( 9 1 ) .
Explanation
Analyze the problem We want to find the probability of picking two green socks in a row from a drawer containing 2 red, 2 blue, 2 yellow, 2 black, and 2 green socks. This is a compound event, meaning the outcome of the first pick affects the probability of the second pick.
Calculate the probability of the first green sock First, let's find the probability of picking a green sock on the first draw. There are 2 green socks out of a total of 10 socks. So, the probability of picking a green sock on the first draw is 10 2 .
Calculate the probability of the second green sock Now, let's find the probability of picking a green sock on the second draw, given that we already picked a green sock on the first draw and did not replace it. After picking one green sock, there is now only 1 green sock left, and the total number of socks remaining is 9. So, the probability of picking a green sock on the second draw is 9 1 .
Multiply the probabilities and find the answer To find the probability of both events happening, we multiply the probabilities of each event: 10 2 × 9 1 . Therefore, the fractions that should be multiplied to find the probability of choosing a pair of green socks are 10 2 and 9 1 .
Examples
This type of probability problem is useful in many real-life scenarios. For example, imagine you are selecting members for a team. If you need two specific roles filled and you are selecting candidates randomly, you can use these probability calculations to determine the likelihood of selecting the right people for those roles. Similarly, in manufacturing, if you are selecting parts for assembly, you can calculate the probability of selecting two defective parts in a row.
To calculate the probability of choosing two green socks from Sal's drawer, multiply the probabilities of picking a green sock on the first draw, which is 10 2 , and on the second draw, which is 9 1 . Therefore, the fractions to multiply are ( 10 2 ) ( 9 1 ) . The answer choice is ( 10 2 ) ( 9 1 ) .
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