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

Answer:

a) 0.084

b) 0.802

c) 15

d) 0.424