QuestionFind the probability that Qian is chosen when 3 out of 12 students are selected. Convert the fraction to a decimal.

Studdy Solution

STEP 1

Assumptions1. There are12 students in the class. Three students will be chosen at random3. We are looking for the probability that Qian will be one of the chosen students

STEP 2

First, we need to find the total number of ways to choose students from12. This is a combination problem, which can be solved using the combination formula.
$C(n, k) = \frac{n!}{k!(n-k)!}$ where n is the total number of students, k is the number of students to be chosen, and "!" denotes factorial.

STEP 3

Now, plug in the given values for n and k to calculate the total number of combinations.
$C(12,3) = \frac{12!}{3!(12-3)!}$

STEP 4

Calculate the total number of combinations.
$C(12,3) = \frac{12!}{3!(12-3)!} =220$

STEP 5

Next, we need to find the number of ways to choose3 students including Qian. This is equivalent to choosing2 students from the remaining11 (since Qian is already chosen), which is also a combination problem.
$C(11,2) = \frac{11!}{2!(11-2)!}$

STEP 6

Calculate the number of combinations including Qian.
$C(11,2) = \frac{11!}{2!(11-2)!} =55$

STEP 7

Now that we have the number of combinations including Qian and the total number of combinations, we can find the probability that Qian will be chosen. This is done by dividing the number of combinations including Qian by the total number of combinations.
$(Qian) = \frac{C(11,2)}{C(12,3)}$

STEP 8

Plug in the values for the number of combinations including Qian and the total number of combinations to calculate the probability.
$(Qian) = \frac{55}{220}$

STEP 9

Calculate the probability that Qian will be chosen.
$(Qian) = \frac{55}{220} =.25$ The probability that Qian will be chosen is.25.

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QuestionFind the probability that Qian is chosen when 3 out of 12 students are selected. Convert the fraction to a decimal.

Studdy Solution

STEP 1

Assumptions1. There are12 students in the class. Three students will be chosen at random3. We are looking for the probability that Qian will be one of the chosen students

STEP 2

First, we need to find the total number of ways to choose students from12. This is a combination problem, which can be solved using the combination formula.
$C(n, k) = \frac{n!}{k!(n-k)!}$ where n is the total number of students, k is the number of students to be chosen, and "!" denotes factorial.

STEP 3

Now, plug in the given values for n and k to calculate the total number of combinations.
$C(12,3) = \frac{12!}{3!(12-3)!}$

STEP 4

Calculate the total number of combinations.
$C(12,3) = \frac{12!}{3!(12-3)!} =220$

STEP 5

Next, we need to find the number of ways to choose3 students including Qian. This is equivalent to choosing2 students from the remaining11 (since Qian is already chosen), which is also a combination problem.
$C(11,2) = \frac{11!}{2!(11-2)!}$

STEP 6

Calculate the number of combinations including Qian.
$C(11,2) = \frac{11!}{2!(11-2)!} =55$

STEP 7

Now that we have the number of combinations including Qian and the total number of combinations, we can find the probability that Qian will be chosen. This is done by dividing the number of combinations including Qian by the total number of combinations.
$(Qian) = \frac{C(11,2)}{C(12,3)}$

STEP 8

Plug in the values for the number of combinations including Qian and the total number of combinations to calculate the probability.
$(Qian) = \frac{55}{220}$

STEP 9

Calculate the probability that Qian will be chosen.
$(Qian) = \frac{55}{220} =.25$ The probability that Qian will be chosen is.25.

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QuestionFind the probability that Qian is chosen when 3 out of 12 students are selected. Convert the fraction to a decimal.

Studdy Solution

STEP 1

Assumptions1. There are12 students in the class. Three students will be chosen at random3. We are looking for the probability that Qian will be one of the chosen students

STEP 2

STEP 3

Now, plug in the given values for n and k to calculate the total number of combinations.C(12,3)=12!3!(12−3)!C(12,3) = \frac{12!}{3!(12-3)!}C(12,3)=3!(12−3)!12!​

STEP 4

Calculate the total number of combinations.C(12,3)=12!3!(12−3)!=220C(12,3) = \frac{12!}{3!(12-3)!} =220C(12,3)=3!(12−3)!12!​=220

STEP 5

Next, we need to find the number of ways to choose3 students including Qian. This is equivalent to choosing2 students from the remaining11 (since Qian is already chosen), which is also a combination problem.C(11,2)=11!2!(11−2)!C(11,2) = \frac{11!}{2!(11-2)!}C(11,2)=2!(11−2)!11!​

STEP 6

Calculate the number of combinations including Qian.C(11,2)=11!2!(11−2)!=55C(11,2) = \frac{11!}{2!(11-2)!} =55C(11,2)=2!(11−2)!11!​=55

STEP 7

STEP 8

Plug in the values for the number of combinations including Qian and the total number of combinations to calculate the probability.(Qian)=55220(Qian) = \frac{55}{220}(Qian)=22055​

STEP 9

Calculate the probability that Qian will be chosen.(Qian)=55220=.25(Qian) = \frac{55}{220} =.25(Qian)=22055​=.25The probability that Qian will be chosen is.25.

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Now, plug in the given values for n and k to calculate the total number of combinations.
$C(12,3) = \frac{12!}{3!(12-3)!}$

Calculate the total number of combinations.
$C(12,3) = \frac{12!}{3!(12-3)!} =220$

Next, we need to find the number of ways to choose3 students including Qian. This is equivalent to choosing2 students from the remaining11 (since Qian is already chosen), which is also a combination problem.
$C(11,2) = \frac{11!}{2!(11-2)!}$

Calculate the number of combinations including Qian.
$C(11,2) = \frac{11!}{2!(11-2)!} =55$

Plug in the values for the number of combinations including Qian and the total number of combinations to calculate the probability.
$(Qian) = \frac{55}{220}$

Calculate the probability that Qian will be chosen.
$(Qian) = \frac{55}{220} =.25$ The probability that Qian will be chosen is.25.