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 including Qian. . Three students will be chosen at random from the12 to lead a group of4.
3. We need to find the probability that Qian will be chosen.

STEP 2

We first need to calculate the total number of ways to choose students from12. This is a combination problem, and we can use 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.
$C(12,3) = \frac{12!}{3!9!} =220$

STEP 5

Next, we need to calculate the number of ways to choose2 students from the remaining11 (excluding Qian). This is because we are assuming Qian is one of the chosen ones.
$C(11,2) = \frac{11!}{2!(11-2)!}$

STEP 6

Calculate the number of combinations excluding Qian.
$C(11,2) = \frac{11!}{2!9!} =55$

STEP 7

Now that we have the number of combinations including Qian and the total number of combinations, we can calculate the probability that Qian will be chosen. The probability is the ratio of the number of desired outcomes to the total number of outcomes.
$(Qian) = \frac{C(11,2)}{C(12,3)}$

STEP 8

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

STEP 9

Calculate the probability.
$(Qian) = \frac{55}{220} = \frac{}{4}$

STEP 10

Finally, we need to convert the fraction to a decimal. We can do this by dividing the numerator by the denominator.
$(Qian) = \frac{}{4} =0.25$ The probability that Qian will be chosen is0.25.

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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 including Qian. . Three students will be chosen at random from the12 to lead a group of4.
3. We need to find the probability that Qian will be chosen.

STEP 2

We first need to calculate the total number of ways to choose students from12. This is a combination problem, and we can use 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.
$C(12,3) = \frac{12!}{3!9!} =220$

STEP 5

Next, we need to calculate the number of ways to choose2 students from the remaining11 (excluding Qian). This is because we are assuming Qian is one of the chosen ones.
$C(11,2) = \frac{11!}{2!(11-2)!}$

STEP 6

Calculate the number of combinations excluding Qian.
$C(11,2) = \frac{11!}{2!9!} =55$

STEP 7

Now that we have the number of combinations including Qian and the total number of combinations, we can calculate the probability that Qian will be chosen. The probability is the ratio of the number of desired outcomes to the total number of outcomes.
$(Qian) = \frac{C(11,2)}{C(12,3)}$

STEP 8

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

STEP 9

Calculate the probability.
$(Qian) = \frac{55}{220} = \frac{}{4}$

STEP 10

Finally, we need to convert the fraction to a decimal. We can do this by dividing the numerator by the denominator.
$(Qian) = \frac{}{4} =0.25$ The probability that Qian will be chosen is0.25.

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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 including Qian. . Three students will be chosen at random from the12 to lead a group of4.3. We need to find the probability that Qian will be chosen.

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!9!=220C(12,3) = \frac{12!}{3!9!} =220C(12,3)=3!9!12!​=220

STEP 5

Next, we need to calculate the number of ways to choose2 students from the remaining11 (excluding Qian). This is because we are assuming Qian is one of the chosen ones.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 excluding Qian.C(11,2)=11!2!9!=55C(11,2) = \frac{11!}{2!9!} =55C(11,2)=2!9!11!​=55

STEP 7

STEP 8

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

STEP 9

Calculate the probability.(Qian)=55220=4(Qian) = \frac{55}{220} = \frac{}{4}(Qian)=22055​=4​

STEP 10

Finally, we need to convert the fraction to a decimal. We can do this by dividing the numerator by the denominator.(Qian)=4=0.25(Qian) = \frac{}{4} =0.25(Qian)=4​=0.25The probability that Qian will be chosen is0.25.

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Assumptions1. There are12 students in the class including Qian. . Three students will be chosen at random from the12 to lead a group of4.
3. We need to find the probability that Qian will be chosen.

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.
$C(12,3) = \frac{12!}{3!9!} =220$

Next, we need to calculate the number of ways to choose2 students from the remaining11 (excluding Qian). This is because we are assuming Qian is one of the chosen ones.
$C(11,2) = \frac{11!}{2!(11-2)!}$

Calculate the number of combinations excluding Qian.
$C(11,2) = \frac{11!}{2!9!} =55$

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

Calculate the probability.
$(Qian) = \frac{55}{220} = \frac{}{4}$

Finally, we need to convert the fraction to a decimal. We can do this by dividing the numerator by the denominator.
$(Qian) = \frac{}{4} =0.25$ The probability that Qian will be chosen is0.25.