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Chef Lloyd's fridge is filled with $7$ milk cartons, $5$ of which contain whole milk. Before he begins to bake, he examines the cartons' expiration dates. If Chef Lloyd checks the expiration dates on $4$ randomly selected cartons, what is the probability that all of them contain whole milk? $\newline$ Write your answer as a decimal rounded to four decimal places.._________

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Find probabilities using combinations and permutations

Full solution

Q. Chef Lloyd's fridge is filled with

7

milk cartons,

5

of which contain whole milk. Before he begins to bake, he examines the cartons' expiration dates. If Chef Lloyd checks the expiration dates on

4

randomly selected cartons, what is the probability that all of them contain whole milk?

\newline

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

Calculate Total Outcomes: Total number of cartons: $7$ $\newline$ Cartons to check: $4$ $\newline$ Calculate total possible outcomes using combinations. $\newline$ Total outcomes: $\binom{7}{4}$
Find Value of $7C4$ : Find the value of $_7C_4$ .
$_7C_4 = \frac{7!}{4!(7-4)!} = \frac{7!}{4!3!} = \frac{7 \times 6 \times 5 \times 4!}{4! \times 3 \times 2 \times 1} = 35$
Calculate Favorable Outcomes: Number of whole milk cartons: $5$ $\newline$ Number of whole milk cartons to check: $4$ $\newline$ Calculate favorable outcomes using combinations. $\newline$ Favorable outcomes: $\binom{5}{4}$
Find Value of $5C4$ : 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 that all selected cartons contain whole milk. $\newline$ Probability = Favorable outcomes / Total possible outcomes $\newline$ = $\frac{5}{35}$ $\newline$ = $0.1429$

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