Question

A florist stocks red roses and white roses. Of these, some have thorns and some do not. Let R be the event that a rose is red, and let T be the event that a rose has thorns. Suppose that the following data has been collected about the roses: 1/4 of the roses have thorns, 1/5 of the red roses do not have thorns, and 3/7 of roses with thorns are white. Consider the experiment of selecting a rose at random. Use the data to find the probability of each of the four events
R ∩ T
R ∩ ~T
~R ∩ T
~R ∩ ~T

Answer 1

For the first one I reasoned that you have a 1/4 chance of selecting a rose with thorns and of that 1/4 3/7 of them would be white. The rest of the 4/7 thorned roses must then be red. So I think P(R ∩ T ) = (1/4)*(4/7) = 1/7

Answer 2

Please let me know if you think I did the first one correctly. I'm stuck on the second one

Answer 3

I think for the next one you would have a 1/7 chance of picking a red rose with thorns and a 4/5 of red roses have thorns then you divide this chance by 4 to get the 1/5 chance there are no thorns P(R ∩ ~T ) = (1/7)*(1/4) = 1/28

Answer 4

I think for the next one you know you have 1/4 chance of getting a rose with thorns and also that 3/7 of them are white so P(~R ∩ T ) = (1/4)*(3/7) = 3/28

Answer 5

first one is right

Answer 6

ok that's a good sign

Answer 7

1/5 of the red roses do not have thorns,, doesn't that mean that R ∩ ~T=1/5 ?

Answer 8

it is late and my head is spinning, but with careful thought we can figure out what proportion of the roses are red and anything else you need

Answer 9

even if 1/5 of the red roses don't have thorns don't we need to consider the probability of picking a white rose as well?

Answer 10

I appreciate it satellite :)

Answer 11

hold on one sec, i will see if i can make it clear

Answer 12

For the last one I thought that if you have a 1/4 chance of picking a rose with thorns then 3/4 must not have thorns. Of this 3/4 you have a 1/28 chance of picking a red rose from the second part. This would give you P(~R ∩ ~T) = (3/4)*(27/28) = 81/112

Answer 13

I don't feel very confident about these

Answer 14

ok my brain is shot, but lets try this, and then we can do it correctly with probabilities
suppose there are 700 roses,
then one fourth of them or 175 have thorns and the rest do not
of those 175, 3/7 are white so there are 75 white roses with thorns, and therefore 100 red roses with thorns
that is 4/5 of the red roses, so there must be 125 red roses all together of which 100 have thorns and 25 to not

Answer 15

i was doing exactly that ^^

Answer 16

so 2n will be 25/700 = 1/28, so thats correct

Answer 17

so we now now the exact composition of these 700 roses
125 red and 575 white
100 red roses have thorn, 25 do not,
75 white roses have thorns,
so 500 do not

now you can answer any question

Answer 18

Yes this is perfect. I think I confirmed all my answers except the last one as well

Answer 19

oh sorry i meant
\[P(R\cap T^c)=\frac{1}{7}\]

Answer 20

you can redo this without the 700 and see that the answers are the same

Answer 21

wouldn't it be 25/700 or 1/28? for P(R∩~T)

Answer 22

yeah i told you my brain was shot

Answer 23

yup, that i already mentioned

Answer 24

last one comes out to be 500/700 = 5/7... ?

Answer 25

last one would be \(\frac{5}{7}\) i think white without thorns
try repeating the same computation without the 700

Answer 26

over and out

Answer 27

thanks again satellite! :D

Answer 28

not red and thorns = white and thorns = 75/175= 3/7

Answer 29

1/7,1/28,3/7,5/7

Answer 30

wouldn't it be 75/700 or 3/28?

Answer 31

but there are only 175 flowers with thorns....

Answer 32

oh, wait, its 75/700,sorry

Answer 33

but you need the probability of choosing from all 700. You aren't guaranteed to get one with thorns

Answer 34

yeah

Answer 35

i also get confused with this

Answer 36

I just don't know how to reason my way through the last one to get 5/7

Answer 37

you have a 3/4 chance to get a rose without thorns

Answer 38

how do you turn that into 5/7?

Answer 39

ah I got it. It's 3/4 chance minus the 1/28 chance you get a red one. So (3/4)-(1/28) = 5/7

Answer 40

or 2/7 with thorns are red, so all others are 1-2/7 = 5/7

Answer 41

yeah that works too

Answer 42

thanks for your help as well hartnn!

Answer 43

even i learned this....its been some time i haven't done such problems....

Answer 44

all thanks to satellite

Answer 45

It's my first time taking statistics, so this is all new to me haha

Answer 46

I still appreciate you taking you time to take a look at this problem anyways

Answer 47

no problem ^_^