Question

A garden has 8 white roses and 11 red roses. One rose is plucked from the garden and then one more rose is plucked, without replacement.

Let the event W = plucking a white rose the first time;
and event R = plucking a red rose the second time.

In two or more complete sentences, explain the meaning of probabilities P(R/W) and P(W/R). Do they have same value?

Answer 1

\(P(W|R)\) means the probability of W given that R has occurred
\(P(R|W)\) means the probability of R given that W has occurred

Answer 2

replace the letter \(W\) by " plucking a white rose the first time"
replace the letter \(R\) by "plucking a red rose the second time. "
to get your complete sentences

Answer 3

okay so R/W equals plucking a red rose the second time/plucking a white rose the first time and W/R equals plucking a white rose the first time/plucking a red rose the second time? @satellite73

Answer 4

lets see if we can say it in idiomatic english rather than math english
that way we can understand it better

Answer 5

not to be picky but \(R|W\) does not "equal" anything
it is an event
it is the event that the second rose chosen was red, given that you know the first rose chosen was white

Answer 6

and W|R is the event that the first rose chosen is white, given that the second rose was red
kinda weird, like going backwards in time

Answer 7

that makes \[P(R|W)\] the probability that the second rose chosen is red, given that the first one was white
now it is a number (not an event)
you can compute this in your head
let me know if you need help doing it

Answer 8

would P(R/W) be P(.61/.42)? @satellite73

Answer 9

hold on

Answer 10

you know you picked a white one first
how many roses are left, and how many are red?

Answer 11

there is 18 left and 11 are red

Answer 12

right (i had to scroll up to check )

Answer 13

that makes
\[P(R|W)\] obvious
especially if we understand what it means
you basically just answered the question

Answer 14

so P(R/W) means that there is 18 roses left and 11 of them are red?
and P(W/R) means that there are 18 roses left and 7 of them are white?

Answer 15

btw \( P(R/W) = P(.61/.42)\) makes no sense whatsoever
\(R|W\) is an event, not a number perhaps you meant
\[P(R|W)=\frac{.61}{.42}\] but that doesn't make sense either
it could be \[\frac{61}{42}\] but that is a number larger than one, so it is not a probability

lets go slow

Answer 16

\[P(R|W)\] is a number
it is the probability that the second one is red, given that the first one is white
you can compute this in your head
you already said there are 18 roses left and 11 red ones
that makes
\[P(R|W)=\frac{11}{18}\] just like that !

Answer 17

Okay so P(W/R) is the probability that the first rose is white
P(W/R) = 8/19

Answer 18

Let the event W = plucking a white rose the first time;

Answer 19

\[P(W)\] is the probability the first rose is white
that is \(\frac{8}{19}\)

Answer 20

we know
\[P(R|W)=\frac{11}{18}\] also
at the moment we do not know
\[P(W|R)\]

Answer 21

it is an odd question if you think about it
\[P(W|R)\] is the probability that the first rose was white if you know the second one was red
like going back in time

Answer 22

they do not have the same value and the second one is more complicated to compute

Answer 23

okay well thank you for trying to help me @satellite73

Answer 24

yw
make sure you understand the different between \(P(A)\) which is a number, and \(A\) which is an event
we can do the last one if you like, but it is a bit complicated
you use something called "baye's formula'

Answer 25

If you dont mind, i would really appreciate your help with the last one @satellite73

Answer 26

A garden has 8 white roses and 11 red roses. One rose is plucked from the garden and then one more rose is plucked, without replacement.

Let the event W = plucking a white rose the first time;
and event R = plucking a red rose the second time.

\[P(W|R)=\frac{P(W\cap R)}{P(R)}\] the numerator is easy
it is
\[P(W\cap R)=\frac{8}{19}\times \frac{11}{18}\]

Answer 27

the denominator is more complicated , it is
\[P(R)=P(R\cap W)+P(R\cap W^c)\]
\[=\frac{8}{11}\times \frac{11}{18}+\frac{11}{19}\times \frac{7}{18}\]

Answer 28

damn typo!!

Answer 29

\[=\frac{8}{19}\times \frac{11}{18}+\frac{11}{19}\times \frac{7}{18}\]

Answer 30

final answer
\[\frac{\frac{8}{19}\times \frac{11}{18}}{\frac{8}{19}\times \frac{11}{18}+\frac{11}{19}\times\frac{7}{18}}\]

Answer 31

i will let you compute those numbers
i will also let you figure out where they came from
gotta run

Answer 32

okay thank you so much! @satellite73