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There are $7$ students performing in the talent show, $5$ of whom will juggle. $\newline$ If $4$ students are randomly chosen to perform during the first section of the show, what is the probability that all of them will juggle? $\newline$ Write your answer as a decimal rounded to four decimal places. $\newline$ ____ $\newline$

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Find probabilities using combinations and permutations

Full solution

Q. There are

7

students performing in the talent show,

5

of whom will juggle.

\newline

If

4

students are randomly chosen to perform during the first section of the show, what is the probability that all of them will juggle?

\newline

Write your answer as a decimal rounded to four decimal places.

\newline

____

\newline

Calculate Probability First Pick: First, calculate the probability of choosing a juggling student on the first pick. There are $5$ jugglers out of $7$ students. $\newline$ Probability for the first pick $= \frac{\text{Number of jugglers}}{\text{Total number of students}} = \frac{5}{7}.$
Calculate Probability Second Pick: Next, calculate the probability of choosing a juggling student on the second pick. After one juggler is picked, there are $4$ jugglers left and $6$ students in total. $\newline$ Probability for the second pick $= \frac{\text{Remaining jugglers}}{\text{Remaining students}} = \frac{4}{6}.$
Calculate Probability Third Pick: Now, calculate the probability of choosing a juggling student on the third pick. There are now $3$ jugglers left and $5$ students in total. $\newline$ Probability for the third pick $= \frac{\text{Remaining jugglers}}{\text{Remaining students}} = \frac{3}{5}.$
Calculate Probability Fourth Pick: Finally, calculate the probability of choosing a juggling student on the fourth pick. There are $2$ jugglers left and $4$ students in total. $\newline$ Probability for the fourth pick $= \frac{\text{Remaining jugglers}}{\text{Remaining students}} = \frac{2}{4}.$
Multiply Probabilities: To find the overall probability that all $4$ chosen students will juggle, multiply the probabilities of each pick together. $\newline$ Overall probability = $(\frac{5}{7}) \times (\frac{4}{6}) \times (\frac{3}{5}) \times (\frac{2}{4})$ .
Simplify Multiplication: Simplify the multiplication of fractions by canceling out common factors. $\newline$ Overall probability = $(\frac{5}{7}) \times (\frac{2}{3}) \times (\frac{3}{5}) \times (\frac{1}{2})$ .
Cancel Common Factors: After canceling out common factors, the remaining multiplication is: $\newline$ Overall probability = $(\frac{1}{7}) \times (\frac{1}{3}) \times (\frac{1}{1}) \times (\frac{1}{2})$ .
Multiply Remaining Fractions: Now, multiply the remaining fractions to get the final probability. $\newline$ Overall probability = $(\frac{1}{7}) \times (\frac{1}{3}) \times (\frac{1}{2}) = \frac{1}{42}$ .

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