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There are $8$ 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

8

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, calculate the probability of choosing a juggling student on the first pick. There are $5$ jugglers out of $8$ students, so the probability is $\frac{5}{8}$ .
Second Pick Probability: Next, if one juggling student is already chosen, there are now $4$ jugglers left out of $7$ students. $\newline$ The probability for the second pick is then $\frac{4}{7}$ .
Third Pick Probability: For the third pick, there are $3$ jugglers left and $6$ students total. $\newline$ The probability is now $\frac{3}{6}$ , which simplifies to $\frac{1}{2}$ .
Fourth Pick Probability: Finally, for the fourth pick, there are $2$ jugglers left out of $5$ students. $\newline$ The probability for the last pick is $\frac{2}{5}$ .
Multiply Probabilities: Multiply all the probabilities together to get the overall probability. $\newline$ $(\frac{5}{8}) \times (\frac{4}{7}) \times (\frac{1}{2}) \times (\frac{2}{5}) = (\frac{5 \times 4 \times 1 \times 2}{8 \times 7 \times 2 \times 5})$
Simplify Multiplication: Simplify the multiplication and division. $\newline$ The $2$ 's and $5$ 's cancel out, leaving $\frac{4}{7} \times \frac{1}{8} = \frac{4}{56}$ , which simplifies to $\frac{1}{14}$ .

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