G C S E Even More Sweets

[kuiperbelt]

A bag contains exactly 7 red sweets, the other remaining sweets are all yellow.
Two sweets are taken from the bag at random and eaten.
The probability of eating 2 yellow sweets is 30%
Find how many sweets were in the bag to begin with?

[ClareTG0ld]

Reverse the previous problem:

there are Y yellow sweets, so initially you can say that the odds of picking a yellow sweet is Y/N, and two in a row is

Y/N*(Y-1)/(N-1) = 0.3 (30%)

rearrange to get

3N(N-1) = 10Y(Y-1).

Too many variables, but we know that N = Y + R(ed) = Y + 7, so we now put this in. Once again you get a quadratic in Y, which should look like

Y^2 - 7 Y - 15 = 0

Now, there is one deliberate typo in the above (but hopefully no accidental ones), so you(r grandson) will have to find it and get the correct equation, but it should be solvable to give an answer for Y, and so for N = Y + 7.

[Prudie]

let number of yellow sweets be Y
So total sweets in bag = 7 +Y
Prob of 1st yellow sweet is y/(7+Y)
Prob of 2nd yellow sweet after the 1st is (Y-1)/(7 +Y-1) = (Y-1)/(6+Y)

We are told that Y/(7+Y) times (Y-1)/(6+Y) =3/10

If you multiply that all out and simplify you will get a quadratic in Y
Y^2 -7Y -18 = 0
That makes Y = 9
So 16 sweets in bag

[ClareTG0ld]

Oh, Prudie, and I even went to the effort of throwing in a typo to check if the student could follow, and you go and spoil it for me :P

Anyhow, I agree.

[Prudie]

I am sorry :-(
TBH I'd rather kuiper specified how he wants these answered. Is it just the answer, is it just hints for his nephew as you've suggested, is it with full working for explanation or is it just for the ABers to try.
I still don't know.

[Gizmonster]

Damn .... I'm waaaay too slow.
When I started working this out, there was only one reply .... I worked it out and refreshed the page to see if there were any more answers and Prudie beat me to it.
Anyway, at least I know my maths is still okay as I got the same formula as Prudie:
Y^2 - 7Y - 18 = 0
Which gives:
(Y + 2)(Y - 9) = 0
.... which gives Y = -2 (which makes no sense) or Y = 9
So total number = 16 :)

[ClareTG0ld]

It's clearly too much to expect kb's grandson to "get" these types of problem from a single example. All the same, the ability to spot that this is, in terms of set up, the same as the previous sweet problem, is a key skill, and one worth developing. To my mind, at least, it isn't developed by having somebody lay out the separate solutions, especially without some amount of input from the student. Where did it become unclear how to solve? Was it, for example, when we were asked about the yellow sweets, rather than the red? Was it somewhere later, when the heavier algebra got harder to work through? Or anywhere in between?

I'd dearly love to have this kind of input. It's a similar question to what prudie's asking, I suppose: *how* are our answers proving useful?

[Prudie]

I suppose everyone is different. I would have found the second half the easy or more fun bit but that's because playing with algebra is fun and you can learn the rules. For me the hard bit is understanding how to put together the probability equations at the start. That requires a slightly different kind of brain and is not easy to lay out black and white rules to help with the solution. It needs some creative thinking,

Years back during one of my teaching practices I was given a 6th form class. The teacher gave me the topic of probability to teach them. Looking back a clever and mean move. I pretty much floundered, It was a horrendous subject to teach.

[kuiperbelt]

Prudie \For me the hard bit is understanding how to put together the probability equations at the start\

You have hit the nail on the head. His algebra and transposition of formula have come on leaps and bounds!! It's when the question is posed in text format he falters not knowing where to begin.

He (We) hope by him doing past exam papers he'll get a question in the final exams which resembles what he did in a past paper.
So please keep the model answers coming!!

\I pretty much floundered, It was a horrendous subject to teach\

Thankyou for sharing this!!

A sincere thankyou to all other posters!!