Raisins in bread

The other day I baked a loaf of bread. To make it sweeter I put in some raisins. I only had 12 raisins to use so I put them all in.

The fresh loaf of bread was lovely. I instantly cut the loaf in 12 slices and I took one slice and made me a sandwich.

What is the probability that ...
(1) ... in my slice there was exactly *one* raisin?
(2) ... there was *none* in my slice?
(3) ... there was exactly *one* raisin in every slice?

If I wanted a probability of 90% that at least *one* raisin should be found in each slice of the lof of bread ...
(4) ... how many raisins should I use in the whole loaf of bread?

Now, I don't have the answers, I have to calculate exactly like you.
Please, give your reasoning along with your answers.

(1) 1/12 * (11/12)^11 (assuming your slice is defined)

(2) (11/12)^12

(3) 11/12 * 10/12 * 9/12 ..... = 11!/(12^11)

(4) You need to define whether you wish there to be a 90% chance that slice A contains at least one raisin and a 90% chance that slice B contains at least one raisin etc.
OR
A 90% chance that every slice contains at least one raisin.

EDIT: It's late, those answers may be wrong.

Originally posted by FabianFnas
The other day I baked a loaf of bread. To make it sweeter I put in some raisins. I only had 12 raisins to use so I put them all in.

The fresh loaf of bread was lovely. I instantly cut the loaf in 12 slices and I took one slice and made me a sandwich.

What is the probability that ...
(1) ... in my slice there was exactly *one* raisin?
(2) ... there ...[text shortened]... rs, I have to calculate exactly like you.
Please, give your reasoning along with your answers.

Did you really bake a loaf of bread? If so, did you do it for the sake of this puzzle, or for the sake of eating it?

Originally posted by ark13
Did you really bake a loaf of bread? If so, did you do it for the sake of this puzzle, or for the sake of eating it?

Myself, I have never actually eaten a puzzle🙂
However, one interesting thing, if you slice the loaf into 12 equal slices, it does seem there would some effect of the volume of the loaf vs the size of each raisin, if they were more than 1/12th the length of the loaf then the probabilities would be effected, as opposed to raisins that were, say, one tenth of a millimeter across, see what I mean? Suppose the raisins were 1/6th the length of the loaf, then you would be only getting pieces of raisins per slice, maybe one total raisin but then you have to talk about percentages of raisins.

Originally posted by XanthosNZ
(3) 11/12 * 10/12 * 9/12 ..... = 11!/(12^11)

I'm sure that first raisin has just gotta be somewhere.

Originally posted by XanthosNZ
(4) You need to define whether you wish there to be a 90% chance that slice A contains at least one raisin and a 90% chance that slice B contains at least one raisin etc.
OR
A 90% chance that every slice contains at least one raisin.

EDIT: It's late, those answers may be wrong.

Yep, it's late alright.

Originally posted by ThudanBlunder
[b]Originally posted by XanthosNZ
(4) You need to define whether you wish there to be a 90% chance that slice A contains at least one raisin and a 90% chance that slice B contains at least one raisin etc.
OR
A 90% chance that every slice contains at least one raisin.

EDIT: It's late, those answers may be wrong.

Yep, it's late alright.[/b]

Those are in fact different despite sounding very similar. If there is a 90% chance of a slice containing a raisin then the chance of all slices containing a raisin is (0.9)^12.

And my logic for question 3 goes as follows:

The first raisin must go into a slice containing zero raisins (as all slices contain zero raisins). Probabilitity 1.
The second raisin goes into a slice containing zero raisins 11/12 ths of the time (as one slice contains a raisin).
The third raisin goes into a slice containing zero raisins 10/12 ths of the time (as two slices contain raisins).
etc.

Originally posted by XanthosNZ
Those are in fact different despite sounding very similar. If there is a 90% chance of a slice containing a raisin then the chance of all slices containing a raisin is (0.9)^12.

And my logic for question 3 goes as follows:

The first raisin must go into a slice containing zero raisins (as all slices contain zero raisins). Probabilitity 1.
The second ...[text shortened]... nto a slice containing zero raisins 10/12 ths of the time (as two slices contain raisins).
etc.

3) Agreed. I thought that you had omitted 12/12 for the first raisin.

4) My take: the number of ways of distributing n raisins among 12 slices such that every slice contains at least one raisin is given by S(n,12), where S(n,k) denotes the Stirling Numbers of the Second Kind.

Total number of ways = 12^n

Therefore we need the least n such that S(n,12) > (0.9)*(12^n)
Solve for n and take the next greatest integer.

http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html

Originally posted by ThudanBlunder
3) I thought that you had omitted 12/12 for the first raisin.

That doesn't sound right. I mean I thought your index was out by 1.

Originally posted by ThudanBlunder
That doesn't sound right. I mean I thought your index was out by 1.

But if I do add that in I'll get the same answer as anything multiplied by 1 is itself.

Originally posted by XanthosNZ
But if I do add that in I'll get the same answer as anything multiplied by 1 is itself.

I know; let's just say I misread your answer and leave it at that. 🙂

Originally posted by FabianFnas
The other day I baked a loaf of bread. To make it sweeter I put in some raisins. I only had 12 raisins to use so I put them all in.

The fresh loaf of bread was lovely. I instantly cut the loaf in 12 slices and I took one slice and made me a sandwich.

What is the probability that ...
(1) ... in my slice there was exactly *one* raisin?
(2) ... there ...[text shortened]... rs, I have to calculate exactly like you.
Please, give your reasoning along with your answers.

You can't answer this question unless you know the size of the loaf and the size of the raisons.

As a minimum, you need to at least know the ratio between the size of the loaf and the size of the raisons.

For example, if the raisons are one cubic centimetre and the loaf is only 20 cubic centimeters, the answer will be much different if the loaf is 100 cubic centimeters.

Originally posted by uzless
You can't answer this question unless you know the size of the loaf and the size of the raisons.

As a minimum, you need to at least know the ratio between the size of the loaf and the size of the raisons.

For example, if the raisons are one cubic centimetre and the loaf is only 20 cubic centimeters, the answer will be much different if the loaf is 100 cubic centimeters.

It is tacitly assumed that the size of a raisin is negligible compared with the size of the loaf.

Originally posted by uzless
You can't answer this question unless you know the size of the loaf and the size of the raisons.

As a minimum, you need to at least know the ratio between the size of the loaf and the size of the raisons.

For example, if the raisons are one cubic centimetre and the loaf is only 20 cubic centimeters, the answer will be much different if the loaf is 100 cubic centimeters.

Ypu can always kind of cheat and ignore the fact that raisins can be cut when slicing, so instead consider each raisin as a single point.

Originally posted by deriver69
Ypu can always kind of cheat and ignore the fact that raisins can be cut when slicing, so instead consider each raisin as a single point.

That was the thrust of my post.