Question

eliana runs a doggy daycare, where 17 dogs are currently enrolled. the collection of dogs includes 13 golden retrievers and 3 bulldogs. if eliana randomly picks the first 4 dogs to give baths to in a certain order, what is the probability that the first 2 dogs are golden retrievers and the last 2 are bulldogs? write your answer as a decimal rounded to four decimal places.

Explanation:

Step1: Calculate probability of first golden retriever

The probability that the first - dog is a golden retriever is $\frac{13}{17}$ since there are 13 golden retrievers out of 17 total dogs.

Step2: Calculate probability of second golden retriever

After picking one golden retriever, there are 12 golden retrievers left out of 16 remaining dogs. So the probability that the second dog is a golden retriever is $\frac{12}{16}$.

Step3: Calculate probability of first bulldog

After picking 2 golden retrievers, there are 3 bulldogs left out of 15 remaining dogs. So the probability that the third dog is a bulldog is $\frac{3}{15}$.

Step4: Calculate probability of second bulldog

After picking one bulldog, there are 2 bulldogs left out of 14 remaining dogs. So the probability that the fourth dog is a bulldog is $\frac{2}{14}$.

Step5: Calculate overall probability

The overall probability is the product of the probabilities of each step. So $P=\frac{13}{17}\times\frac{12}{16}\times\frac{3}{15}\times\frac{2}{14}=\frac{13\times12\times3\times2}{17\times16\times15\times14}=\frac{936}{57120}\approx0.0164$.

Answer:

0.0164