Question

tyler and some of his friends are going rock climbing this weekend. in preparation, tyler purchased 34 sports drinks to bring, including 19 orange flavored drinks. if tyler randomly chooses to place 31 drinks in the red cooler, what is the probability that exactly 18 of the chosen sports drinks are orange flavored? write your answer as a decimal rounded to four decimal places.

Explanation:

Step1: Calculate total combinations

The total number of ways to choose 31 drinks out of 34 is given by the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 34$ and $r=31$. So $C(34,31)=C(34,3)=\frac{34!}{3!(34 - 3)!}=\frac{34\times33\times32}{3\times2\times1}=5984$.

Step2: Calculate favorable combinations

The number of ways to choose 18 orange - flavored drinks out of 19 is $C(19,18)=\frac{19!}{18!(19 - 18)!}=19$. The number of ways to choose the remaining $31-18 = 13$ non - orange drinks out of $34 - 19=15$ non - orange drinks is $C(15,13)=C(15,2)=\frac{15!}{2!(15 - 2)!}=\frac{15\times14}{2\times1}=105$. The number of favorable combinations is $C(19,18)\times C(15,13)=19\times105 = 1995$.

Step3: Calculate probability

The probability $P$ is the number of favorable combinations divided by the total number of combinations. So $P=\frac{1995}{5984}\approx0.3334$.

Answer:

$0.3334$