Question

8. Suppose in a certain breed of dog, the color of fur can either be tan or black. Eighty-five percent of the time, a puppy will be born with tan fur, while 15% of the time, the puppy will have

black fur. Suppose in a future litter, six puppies will be born.
a. Are the events having tan fur and having black fur independent? Explain.
b. What is the probability that one puppy in the litter will have black fur and another puppy will have tan fur?
c. What is the probability that all six puppies will have tan fur?
d. Is it likely for three out of the six puppies to be born with black fur? Justify mathematically.

Answer 1

a) On this case the two events are independent, because if one puppy is black the probability that the next one would be black or tan fur is not affected by the first outcome. On math terms we have this:

`P(T|B) = P(T) = 0.85`

`P(B|T) =P(B) =0.15`

b) On this case since the order no matter the probability would be:

`P(B and T) = 0.15*0.85= 0.1275`

And the two events are independent.

c) For this case the probability that all six puppies will have tan fur is:

`P(6 Tan fur) = (0.85^6) = 0.377`

d) On this case the probability that 3 puppies would be black fur means that the other 3 are tan fur. And the probability would be given by:

`(0.15^3) (0.85^3) = 0.00207`

So on this case is very unlikely since we have just about 0.2% of possibility.

Explanation:

Notation

T= a pupy is tan fur, P(T) = 0.85

B= a puppy is black fur P(B) = 0.15

a. Are the events having tan fur and having black fur independent? Explain.

On this case the two events are independent, because if one puppy is black the probability that the next one would be black or tan fur is not affected by the first outcome. On math terms we have this:

`P(T|B) = P(T) = 0.85`

`P(B|T) =P(B) =0.15`

b. What is the probability that one puppy in the litter will have black fur and another puppy will have tan fur?

On this case since the order no matter the probability would be:

`P(B and T) = 0.15*0.85= 0.1275`

And the two events are independent.

c. What is the probability that all six puppies will have tan fur?

For this case the probability that all six puppies will have tan fur is:

`P(6 Tan fur) = (0.85^6) = 0.377`

d. Is it likely for three out of the six puppies to be born with black fur? Justify mathematically.

On this case the probability that 3 puppies would be black fur means that the other 3 are tan fur. And the probability would be given by:

`(0.15^3) (0.85^3) = 0.00207`

So on this case is very unlikely since we have just about 0.2% of possibility.