Question

A certain flower can be white or blue and may or may not have fragrance. Suppose blue flowers are extremely rare and that the rate of blue flowers in the population is 1/400 flowers. Consider the following data.

Fragrant Not Fragrant Total Sampled
White 500 500 1000
Blue 2 8 10

If we happen to smell a randomly selected fragrant flower, what is the probability it is blue?

Answer 1

Rate of white flowers in population = 399/400 flowers = 798/800 flowers
Rate of fragrant white flowers in population =
\[\frac{798}{800}\times \frac{1}{2}=\frac{399}{800}\]
Rate of blue flowers in population = 1/400 flowers = 2/800 flowers
Rate of fragrant blue flowers in population =
\[\frac{2}{800}\times \frac{1}{5}=\frac{2}{5}\times \frac{1}{800}\]
Rate of fragrant white plus blue flowers in population = (399 + 2/5)/800 flowers
\[P(blue\ fragrant)=\frac{\frac{2}{5}}{399+\frac{2}{5}}\]

Answer 2

how did you come up with the 798/800 ?

Answer 3

i mean the 399/400*

Answer 4

It is given that the rate of blue flowers in the population is 1/400 flowers. Therefore on average out of 400 flowers 1 is blue and 400 - 1 = 399 are white. So the rate of white flowers (fragrant + not fragrant) in the population is 399/400.

Answer 5

oh okay, that makes sense. thank you!

Answer 6

You're welcome. Please let me know if you need more clarification.

Answer 7

do you need to double the rate, ie. 399/400 to 798/800 ?

Answer 8

The rate is the same when the numerator and denominator are both multiplied by 2.
399/400 = 798/800.
I increased the reference quantity from 'per 400 flowers' to 'per 800 flowers' to avoid having 199.5/400 at the next step. It looks better to see 399/800, although:
199.5/400 = 399/800.

Answer 9

oh okay, thank you

Answer 10

You're welcome :)