Question

Every day dogs come to the dog park to frolic. Age of dogs: 24% of the dogs that come are young, (the rest are seniors). Poodles are more popular today than a few years ago, so around 35% of young dogs are Poodles, while only 15% of senior dogs are Poodles.Create a tree diagram for this situationWhat is the total probability that any given dog at the dog park is a young poodle?What is the probability that for any given dog they are not a Poodle?If there are 130 dogs at the park today, how many of them are adult poodles?A dog is picked at random, given that it is a poodle, what is the chance it is a young dog?

Answer 1

We will construct a tree diagram with the information given:

• 24% of the dogs are young, so 76% are seniors.

,

• Of the young dogs, 35% are poodles.

,

• Of the senior dogs, 15% are poodles.

We can draw the diagram as:

a) We have to calculate the probability that a given dog is a young poodle.

We can calculate it as:

$\begin{gathered} P(Y\cap P)=P(P|Y)\cdot P(Y) \\ P(Y\cap P)=0.35\cdot0.24 \\ P(Y\cap P)=0.084 \end{gathered}$

b) We can calculate the probability that any given dog is not a puddle as:

$\begin{gathered} P(\text{not P})=1-P(P) \\ P(\text{not P})=1-\lbrack P(P|Y)\cdot P(P)+P(P|S)\cdot P(S)\rbrack \\ P(\text{not P})=1-\lbrack0.35\cdot0.24+0.15\cdot0.76\rbrack \\ P(\text{not P})=1-(0.084+0.114) \\ P(\text{not P})=1-0.198 \\ P(\text{not P})=0.802 \end{gathered}$

c) If there are 130 dogs, we can use the proportion of adult poodles to calculate how many there are:

$P(S\cap P)=P(P|S)\cdot P(S)=0.15\cdot0.76=0.114$

We can multiply this proportion by 130 and get the number of expected senior poodles:

$X=N\cdot P(S\cap P)=130\cdot0.114=14.82\approx15$

d) Now, we have to calculate the probability that any given dog is a young dog, given that is a poodle. We can do this as:

$\begin{gathered} P(Y|P)=\frac{P(Y\cap P)}{P(P)} \\ P(Y|P)=\frac{P(P|Y)\cdot P(Y)}{P(P|Y)\cdot P(Y)+P(P|S)\cdot P(S)} \\ P(Y|P)=\frac{0.35\cdot0.24}{0.35\cdot0.24+0.15\cdot0.76} \\ P(Y|P)=\frac{0.084}{0.084+0.114} \\ P(Y|P)=\frac{0.084}{0.198} \\ P(Y|P)\approx0.424 \end{gathered}$

Answer:

a) 0.084

b) 0.802

c) 15

d) 0.424