For this problem, assume that three students are chosen from 3 freshmen, 7 sophomores, and 5 juniors. What is the probability of selecting 2 freshmen and 1 junior given that at least one freshman will be selected?

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For this problem, assume that three students are chosen from 3 freshmen, 7 sophomores, and 5 juniors.  What is the probability of selecting 2 freshmen and 1 junior given that at least one freshman will be selected?

falhiblesw · Accepted Answer

Step 1  Given: Number of Freshmen, Sophomores &amp;amp; juniors are 3, 7 &amp;amp; 5 respectively.  To find: Probability of selecting 2 freshmen, 1 junior given that at least one freshman will be selected.   Total number of students is equal to the sum of number of freshmen, sophomores and juniorsTotal number of students 3+7+5  =15 The formula which gives us the total number of ways of forming group of r objects, out of n objects is:  C(n,r)=n!r!(n−r)  Step 2  Three students are selected from 3 freshmen, 7 sophomores and 5 juniors, Such that the at least 1 freshman is selected  The selection of one freshman can be done in  P(3, 1) ways  P(3,1)=3!(3−1)!=3  Selection of one freshman can be done in 3 ways  The selection of remaining two students from  2 freshmen, 7 sophomores and 5 juniors.  Number of ways of selection two students out of (2+7+5)=14 students  C(14,2)=14!2!(14−2)!  C(14,2)=7×13  C(14,2)=91  Thus total number of possible ways of selection three students are =3×91=273  Step 3  Selection of two freshman from three freshmen is done in C(3, 2) ways  C(3,2)=3!2!(3−2)!  C(3, 2)=3  Thus, the selection of two freshman and one junior can be done in C(3, 2)×3 ways  C(3, 2)×3=3×3  C(3, 2)×3=9  The selection of two freshman and one junior can be done in 9 ways.  Step 4  Probability of selection of two freshman and one junior out of the total number of possible ways of selection three students such that at least one freshman is selected.  Number of favorable outcomes =9  Total number of outcomes =273  Probability =Number of favorable outcomestotal number of outcomes  Probability =9273  Probability =0.032  Answer:  Probability of selection of two freshman and one junior given that at least one freshman is selected is 0.032.

Jazz Frenia · Accepted Answer

Step 1: Let&#039;s define the following events:A: Selecting 2 freshmen and 1 junior.B: At least one freshman is selected.We want to find the probability of event A given event B, denoted as P(A|B).To calculate P(A|B), we need to find two probabilities: P(A) and P(B).The probability of event A can be calculated as follows:P(A)=number&amp;nbsp;of&amp;nbsp;ways&amp;nbsp;to&amp;nbsp;select&amp;nbsp;2&amp;nbsp;freshmen&amp;nbsp;and&amp;nbsp;1&amp;nbsp;juniortotal&amp;nbsp;number&amp;nbsp;of&amp;nbsp;possible&amp;nbsp;selectionsThe number of ways to select 2 freshmen and 1 junior can be calculated by multiplying the number of ways to choose 2 freshmen from 3 freshmen and the number of ways to choose 1 junior from 5 juniors:{number&amp;nbsp;of&amp;nbsp;ways&amp;nbsp;to&amp;nbsp;select&amp;nbsp;2&amp;nbsp;freshmen&amp;nbsp;and&amp;nbsp;1&amp;nbsp;junior}=(32)×(51)The total number of possible selections can be calculated by choosing 3 students from the total number of students (3 freshmen + 7 sophomores + 5 juniors):{total&amp;nbsp;number&amp;nbsp;of&amp;nbsp;possible&amp;nbsp;selections}=(3+7+53)Substituting these values into the formula, we get:P(A)=(32)×(51)(3+7+53)Step 2: Now, let&#039;s calculate the probability of event B:P(B)=1−P({no&amp;nbsp;freshman&amp;nbsp;is&amp;nbsp;selected})The probability of no freshman being selected can be calculated by choosing 3 students from the combined group of sophomores and juniors:P({no&amp;nbsp;freshman&amp;nbsp;is&amp;nbsp;selected})=(7+53)(3+7+53)Finally, we can calculate P(A|B) using the formula for conditional probability:P(A|B)=P(A∩B)P(B)Simplifying further:P(A|B)=P(A)P(B)Substituting the values we calculated earlier:P(A|B)=(32)×(51)(3+7+53)1−(7+53)(3+7+53)Solving this expression will give us the desired probability.

xleb123 · Accepted Answer

The total number of possible selections can be calculated by choosing 3 students from the total number of students available. In this case, we have 3 freshmen, 7 sophomores, and 5 juniors. Therefore, the total number of possible selections is given by the binomial coefficient:(3+7+53)=15!3!·7!·5!Next, let&#039;s determine the number of favorable selections, which is the number of ways we can choose 2 freshmen and 1 junior, given that at least one freshman will be selected. We can consider two cases: selecting 2 freshmen and 1 junior, or selecting all 3 freshmen.Case 1: Selecting 2 freshmen and 1 juniorThe number of ways to select 2 freshmen from 3 is given by the binomial coefficient:(32)=3!2!·1!The number of ways to select 1 junior from 5 is given by:(51)=5!1!·4!So the total number of favorable selections for this case is:(32)·(51)Case 2: Selecting all 3 freshmenIn this case, we have only one favorable selection.Therefore, the total number of favorable selections is:(32)·(51)+1Finally, we can calculate the probability by dividing the number of favorable selections by the total number of possible selections:P({2&amp;nbsp;freshmen&amp;nbsp;and&amp;nbsp;1&amp;nbsp;junior}|{at&amp;nbsp;least&amp;nbsp;one&amp;nbsp;freshman})=(32)·(51)+1(3+7+53)

Andre BalkonE · Accepted Answer

Answer:37!5!2!Explanation:F: At least one freshman is selected.J: Exactly one junior is selected.We need to find the probability of event J given event F, denoted as P(J|F). By using the formula for conditional probability, we have:P(J|F)=P(J∩F)P(F)To find P(J∩F), we can consider the following:1. Choose 2 freshmen from 3 freshmen: (32)2. Choose 1 junior from 5 juniors: (51)3. Choose 0 sophomores from 7 sophomores: (70)Thus, the number of ways to select 2 freshmen, 1 junior, and 0 sophomores is (32)×(51)×(70).To find P(F), we can calculate the probability of selecting at least one freshman. This can be done by considering the complement event, i.e., the event of selecting no freshmen. The number of ways to select 0 freshmen, 1 junior, and 2 sophomores is (30)×(51)×(72).Now, let&#039;s substitute these values into the formula for conditional probability:P(J|F)=(32)×(51)×(70)(30)×(51)×(72)Calculating the combinations, we get:P(J|F)=3!2!(3−2)!×5!1!(5−1)!×7!0!(7−0)!3!0!(3−0)!×5!1!(5−1)!×7!2!(7−2)!Simplifying further, we have:P(J|F)=3!2!×5!4!×13!3!×5!4!×7!5!2!Finally, simplifying the expression, we get:P(J|F)=37!5!2!Therefore, the probability of selecting 2 freshmen and 1 junior given that at least one freshman will be selected is 37!5!2!.

For this problem, assume that three students are chosen from 3 freshmen, 7 sophomores, and 5 juniors. What is the probability of selecting 2 freshmen and 1 junior given that at least one freshman will be selected?

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