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Tony and Whitney made a list of $9$ psychology experiments they want to try at home. Each experiment tackles one specific psychology topic. $7$ of the experiments involve music and memory. $\newline$ If Tony and Whitney randomly choose $6$ experiments to try over the summer, what is the probability that all of them involve music and memory? $\newline$ 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. Tony and Whitney made a list of

9

psychology experiments they want to try at home. Each experiment tackles one specific psychology topic.

7

of the experiments involve music and memory.

\newline

If Tony and Whitney randomly choose

6

experiments to try over the summer, what is the probability that all of them involve music and memory?

\newline

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

\newline

____

\newline

Calculate Total Ways: First, calculate the total number of ways to choose $6$ experiments from $9$ . This is a combination problem, so use the formula for combinations: $C(n, k) = \frac{n!}{k!(n-k)!}$ . Calculate $C(9, 6) = \frac{9!}{(6!(9-6)!)} = \frac{9!}{(6!3!)} = \frac{(9\times8\times7)}{(3\times2\times1)}$ .
Calculate Music and Memory: Now, calculate the total number of ways to choose $6$ experiments that all involve music and memory. $\newline$ Since there are $7$ experiments involving music and memory, calculate $C(7, 6) = \frac{7!}{(6!(7-6)!)} = \frac{7!}{(6!1!)} = \frac{7}{1}$ .
Find Probability: To find the probability, divide the number of favorable outcomes by the total number of possible outcomes. $\newline$ Probability = $\frac{C(7, 6)}{C(9, 6)} = \frac{7}{9\times8\times7 / (3\times2\times1)}$ .
Simplify Calculation: Simplify the probability calculation. $\newline$ Probability = $\frac{7}{(9\times8\times7 / 6)} = \frac{7}{(9\times8\times7 / 6)} = \frac{7}{(3\times8\times7)} = \frac{1}{(3\times8)} = \frac{1}{24}$ .
Convert to Decimal: Convert the probability to a decimal rounded to four decimal places. $\newline$ Probability (decimal) = $\frac{1}{24} \approx 0.0417$ .

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