Just enjoyed a bit of sunshine along with a nice cuppa and biscuit, watching the garden grow and began wondering …
I had a tin containing 5 chocolate biscuits and one ginger biscuit. I took out one biscuit at random and ate it. Delicious !
I then took out a second biscuit, also at random, and ate that one too.
What was the probability that this second biscuit was chocolate ?
I don’t mean to be a spoiler, but I have to ask–did you mean to imply that the first biscuit was chocolate?
Ooops!! Thank you JR. What a gentleman.
I have modified the text to (hopefully) remove that implication.
Note to self (not for the first time) “must try harder !”
Perhaps I should have posted both versions - New Teaser “Spot the difference !”
I also spotted the difference
I preferred your comment when you also had the quotation marks
Fairly easy … is there no way to send a private/direct message so as not to post a spolier.
I just usually give cryptic answers if I want to avoid giving spoilers. They are sometimes teasers in their own right!
I thought that also, but then decided it was too easy, so I removed them.
As Ravvie has said, cryptic answers usually work quite well.
In this instance, (I think) the solutions to both versions can be presented as vulgar fractions. So I might suggest that a ‘D’ an ‘F’ and a couple of “E’s” would be useful ?
Perhaps JR could post one of the solutions (first-come etc) then you could post the other solution.
However, i’m confident that all three of you know both solutions, well-done !
While we await proper answers, here’s my cryptic effort (Don - no offence intended):
Don says a lot but tells us nothing. No change there.
No mathematical or other non-cryptic solutions yet, so maybe I should explain my cryptic answer.
Unlike exam questions (where all information is usually expected to be relevant), teasers often have unnecessary “noise”, which is intended to mask any key information. That didn’t catch out jrhardee who spotted the word “also”, which was the only piece of info that was relevant. Don removed this in the edit which changes things entirely. Two puzzles for the price of one.
Stripping out the noise, the modified puzzle is now: “Given 5 chocolate biscuits and 1 ginger biscuit, what is the chance that the second one is chocolate?” Answer is that each position is just as likely as any other. It might simplify things if we just consider the ginger biscuit instead.
A mathematical answer would be more robust and I think quicker to explain. It’s just that my preferred approach to such problem solving is lateral thinking (or gut feel) to simplify, then justified logic, then maths in third place.
I feel it’s time to put the lid on the biscuit tin and it’s associated teaser(s)
I’ve reposted the teaser in the form I initially intended so that we can focus on that version first.
As Ravvie hinted, there is more than one way to tackle some of these probability teasers. I tend to use “tree diagrams” - at least that’s what I think they are called. But even then, as Ravvie hinted, you can either focus on the chocolate biscuits or the ginger biscuit.
Three possibilities:
The first is chocolate and the second is ginger, 5/6 x 1/5, or 17%.
If the first one is ginger, the second must be chocolate, 1/6 x 1, or 17%.
The first and the second are chocolate, 5/6 x 4/5, or 67%.
All of the possibilities add up to 100%, so I think I got there. The likelihood of the second one being chocolate is 17% + 67% or ~84%, ignoring rounding.
When I did the maths as a check that my logic was sound, I followed the same approach. Getting it to add up to 1 nails it of course.
Mrs R went with the ginger route. It can only be ginger if the first was chocolate, so the probability for ginger at the second round is 5/6 x 1/5 = 1/6. She concluded 5/6 for chocolate. She teaches probability and often uses the neat trick of looking at the thing we don’t want then subtracting from 1.
In my loose logic approach, I went with the ginger biscuit having a 1 in 6 chance of being picked at any stage, and as we are told nothing more, this remains the case.
If Don had asked a harder question, such as he chose and ate a third, fourth (or more) delicious biscuit then my answer would still be 1/6 for ginger and 5/6 for chocolate for the final biscuit eaten. As Don says, a probability tree diagram would show all the possibilities but such a maths approach could take a while especially if it was a large tin with various biscuits.
You got there ! Well done !
Just to avoid rounding …
… your first line shows ginger as 1/6. This tells us that the other possibility (chocolate) has to be 5/6 (83.333…)
Or, as you did, lines 2 and 3 show chocolate as 1/6 + 4/6 = 5/6
Either way - well done !
I did seem to do a bit of unnecessary extra work, didn’t I?
Not really, it was just a rigorous validation.
Thought I might post my Probability Tree, but extended to illustrate Ravvie’s observation that, not only will the second biscuit have a probability 5/6 of being chocolate, but so will the 3rd, 4th etc.
PS. As is often the case, best to click bottom right corner of the picture to enlarge and make it readable.
ALSO !!
What a mess I created by including the word “also” in the initial text of the biscuit teaser !
Thanks to all who commented, I felt I had to remove the word to avoid potential ambiguity. The following conversation with Mrs D might illustrate the ambiguity I wanted to avoid …
Mrs D asked me “when do we place our bets ?”
I answered “right at the start, before any biscuits have been selected or eaten !”
“Oh !” she said “I thought you were inviting bets, after the first biscuit, (which I knew was chocolate), had been selected and eaten”… She continued … “So, I assumed I was now working out the probability of selecting a chocolate biscuit, knowing there were four chocolate and one ginger biscuits left in the tin !”
By removing that word “also”, rendered the teaser a bit less ambiguous - IMHO.
Note to Self “must try harder next time !”
Hopefully some of us enjoyed the teaser and the muddle. And especially Ravvie’s contribution about the third, fourth or fifth biscuit.
