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John has $7$ buttons, $5$ of which are blue. $\newline$ If John randomly picks $4$ buttons, what is the probability that all of them are blue? $\newline$ 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. John has

7

buttons,

5

of which are blue.

\newline

If John randomly picks

4

buttons, what is the probability that all of them are blue?

\newline

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

\newline

____

\newline

Total buttons calculation: Total buttons: $7$ $\newline$ Blue buttons: $5$ $\newline$ Buttons to pick: $4$ $\newline$ Calculate total possible outcomes using combinations. $\newline$ Total outcomes: $\binom{7}{4}$
Calculate _{$\newline$ $7$ $\newline$}C_{$\newline$ $4$ $\newline$}: Find the value of _{$\newline$ $7$ $\newline$}C_{$\newline$ $4$ $\newline$}. $\newline$ $\binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4!3!} = \frac{7 \times 6 \times 5 \times 4!}{4! \times 3 \times 2 \times 1} = 35$
Favorable outcomes calculation: Calculate favorable outcomes where all picked buttons are blue. $\newline$ Favorable outcomes: $\binom{5}{4}$
Calculate ${}_5C_4$ : Find the value of ${}_5C_4$ . ${}_5C_4 = \frac{5!}{4!(5-4)!} = \frac{5!}{4!1!} = \frac{5 \times 4!}{4! \times 1} = 5$
Calculate probability: Calculate the probability. $\newline$ Probability = Favorable outcomes / Total outcomes $\newline$ = $\frac{5}{35}$ $\newline$ = $0.1429$ after rounding to four decimal places.

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