Probability

[unkempt hawk]

Sample word problem while i was studying probabilities, having trouble finding the right answer since it has to be a whole number, i tried all possible cases, did the algebra all the way around, did trial and error on x, as well as the quadratic and either i end up with a negative discriminant or a decimal number. I need help interpreting the probability of each cases in this word problem

[tidal thistle]

Ok imagine the scenario, Elliot has 5 of the x counters to be facing red, then how many are facing green? Now there must be two different cases. First case is that if Elliot flips two different counters once each. You can see that if both were originally facing red, then there would be 3 showing red at the end, however if both were originally facing green, then there would be 7 showing red. In between however, if Elliot flips a red facing counter, it becomes green and a green facing counter , it becomes red, so there’s still 5 red facing counters. Second case is that if Elliot flips the same counter twice. Notice that flipping the same counter twice doesn’t change the face of the counter, so we can flip any counter twice. Add the two cases together should give you the result. I got a quadratic equation with an integer answer

[unkempt hawk]

Do share your solution, most of my tries seem to end with a non discrete answer and/or doesn't match up well with given probability 19/32

I really appreciate your help :))

I possibly may have been interpreting the probabilities for some cases wrong

[tidal thistle]

Did you try the first case? Tell me what probability you got for that, and I can correct you

[tidal thistle]

All good. I’m happy to help

[tidal thistle]

There’s only two cases btw

[unkempt hawk]

Ill update you with a photo of my solution later

[tidal thistle]

Sure

[unkempt hawk]

Ohh, my solution had 3 cases

1. flip to red, flip same card back to green
2. flip to green, flip same card back to red
3. flip a green, flip a red and vice versa

[tidal thistle]

The first two cases can be put together into one case all together. Because if you’re flipping any counter twice, it shows the same face, therefore preserving the same number of red faces originally had

[unkempt hawk]

Yea but the problem is that the the amount of cards with the red faces up is 5, the cards with green faces up is x - 5

The amount of cards is unknown

[tidal thistle]

Yes that’s correct

[tidal thistle]

The total number of counters is the sum of the two. That’s because a counter has a red face and a green face and at a moment is showing one of the two faces