Erica and her stepmother wanted to start volunteering together. They found 7 opportunities online, 5 of which involved working with the elderly. If they randomly chose to apply to 4 of the opportunities, what is the probability that all of them involve working with the elderly? Write your answer as a decimal rounded to four decimal places. ____

Total Outcomes Calculation: Total number of opportunities: 7 Opportunities to choose: 4 Calculate total possible outcomes using combinations. Total outcomes: 7C4Calculate Value of 7C4: Find the value of 7C4. 7C4 = (7!) / (4!(7-4)!) =(7!) / (4!3!) =(7 * 6 * 5 * 4!) / (4! * 3 * 2 * 1) = 35Favorable Outcomes Calculation: Number of opportunities working with the elderly: 5 Number of chosen opportunities with the elderly: 4 Calculate favorable outcomes using combinations. Favorable outcomes: 5C4Calculate Value of 5C4: Find the value of 5C4. 5C4 = (5!) / (4!(5-4)!) =(5!) / (4!1!) = (5 * 4!) / (4! * 1) = 5Calculate Probability: Calculate the probability that all chosen opportunities involve working with the elderly. Probability = Favorable outcomes / Total possible outcomes = 5 / 35 = 0.1429 when rounded to four decimal places.

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Erica and her stepmother wanted to start volunteering together. They found $7$ opportunities online, $5$ of which involved working with the elderly. $\newline$ If they randomly chose to apply to $4$ of the opportunities, what is the probability that all of them involve working with the elderly? $\newline$ Write your answer as a decimal rounded to four decimal places. $\newline$ ____ $\newline$

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Math Problems

Algebra 2

Find probabilities using combinations and permutations

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Q. Erica and her stepmother wanted to start volunteering together. They found

7

opportunities online,

5

of which involved working with the elderly.

\newline

If they randomly chose to apply to

4

of the opportunities, what is the probability that all of them involve working with the elderly?

\newline

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

\newline

____

\newline

Total Outcomes Calculation: Total number of opportunities: $7$ $\newline$ Opportunities to choose: $4$ $\newline$ Calculate total possible outcomes using combinations. $\newline$ Total outcomes: $\binom{7}{4}$
Calculate Value of $7C4$ : Find the value of $_7C_4$ .
$_7C_4 = \frac{7!}{4!(7-4)!} = \frac{7!}{4!3!} = \frac{7 \times 6 \times 5 \times 4!}{4! \times 3 \times 2 \times 1} = 35$
Favorable Outcomes Calculation: Number of opportunities working with the elderly: $5$ $\newline$ Number of chosen opportunities with the elderly: $4$ $\newline$ Calculate favorable outcomes using combinations. $\newline$ Favorable outcomes: $\binom{5}{4}$
Calculate Value of $5C4$ : Find the value of $_5C_4$ . $\newline$ $_5C_4$ = $\frac{5!}{4!(5-4)!}$ $\newline$ = $\frac{5!}{4!1!}$ $\newline$ = $\frac{5 \times 4!}{4! \times 1}$ $\newline$ = $5$
Calculate Probability: Calculate the probability that all chosen opportunities involve working with the elderly. $\newline$ Probability = Favorable outcomes / Total possible outcomes $\newline$ = $\frac{5}{35}$ $\newline$ = $0.1429$ when rounded to four decimal places.