A probability problem by Clara Blackstone

Probability Level 2

In my fridge's shelves there are 10 condiments that are good for breakfasts, 8 that are good for lunch/dinner, and 12 that are good for dessert.

Of all of the condiments, exactly half of those that are good for breakfast are also good for lunch/dinner. And every condiment is either good for desert or good for both lunch/dinner and breakfast.

Given all of this, what is the difference between the maximum number of condiments that I might have in my fridge and minimum number of condiments that I might have in my fridge?

0 Given what we know, there could be infinitely many condiments, so "infinity!" 18 4 5

1 solution

Clara Blackstone
Oct 12, 2015

"Of all of the condiments, exactly half of those that are good for breakfast are also good for lunch/dinner and all of the condiments that are not good for desert are among them." Translates to: "if it's not good for desert, then it's good for both lunch/dinner and breakfast. Given that, here's what we can deduce about the condiment-type distribution:

So, either there are as many as 4 condiments that are good for all meals, in which case there are 13 condiments altogether; OR there are as few as 1 condiments that are good for all meals, in which case there are 17 condiments altogether. So the difference between this min and max is $\fbox{4}$ .

I took "And every condiment is either good for desert or good for both lunch/dinner and breakfast" to be purely binary. That is, there can't be a condiment that's BOTH good for desert AND good for (both lunch/dinner and breakfast) i.e. the intersection between all three was zero. That's why I put my answer for the difference as zero.

Sandra Tan - 4 years, 9 months ago

Me too. :)

Chris Leonard - 3 years ago

"either good for desert or good for both lunch/dinner and breakfast."

Either / or means no intersection. Therefore the answer is 0.

Michael Boyd - 4 years, 6 months ago

take the above statement :"either good for desert or good for both lunch/dinner and breakfast." to be true then, "whenever not good for desert then good for both lunch and breakfast" and, "whenever good for desert it may or may not be good for both lunch and dinner".

Shubham Kumar - 2 years, 5 months ago

On what justification? If "Either A or B" means one or the other but not both, then "Not A" implies "B" and "A" implies "not B", not "B or not B"! That's just plain plain logical A or B.

English is ambiguous. "A or B" particularly so. But normally "either A or B" is used to mean exclusive rather than inclusive or. Using it in a logical puzzle and then claiming it means inclusive is at best unwittingly misleading, and at worst dam right disingenuous.

Michael Boyd - 2 years, 1 month ago

I arranged the condiments like this: BD BD BD BD BD (all of these can be D) BL BL BL BL BL (none of these can be D) LD LD LD (all of these can be D) D D D D (these 4 are left over) but got no Max/Min, just 17 condiments. I chose 4 and got the "right" answer but I also agree the difference is 0 after reading other comments.

Scott Broughton - 3 years, 8 months ago

I believe the least count for all meals should be 0, not 1. That is the number in blue in the intersection of all three meals.

Tom Capizzi - 4 years, 6 months ago

Same as Michael Boyd

Khushi Mehta - 4 years, 5 months ago

The question was written in English and the English word or is the logical xor. If you don't want to play it like that it should be noted

Joe Ridge - 3 years, 2 months ago

The English use or "or" is very ambiguous. "You can reach us by phone or email" does not preclude both. "Either or" is usually logical xor. "You can reach us either by phone or email" sounds wrong because obviously you could do both.

Michael Boyd - 2 years, 1 month ago

I am very sorry that i don't understand. :-) :

--->...10 condiments that are good for breakfasts, 8 that are good for lunch/dinner....

...yeah, but that's IMPOSSIBLE because

--->... EVERY condiment is EITHER good for desert OR good for BOTH lunch/dinner and breakfast.

So we are talking EVERY condiment. Every single one can be good for dessert, or, the only presented alternative is that it is good for both other options.

Since we are talking EVERY -> there is no condiment that does not fall into one of the two presented categories.

For example the fact that the number of condiments that are good for breakfast is different from the ones good for launch/dinner.. I personally find this directly contradictory... What am I missing?

Hideki Yamamoto - 2 years, 11 months ago

You're missing the logical step that those that are good for breakfast but not lunch (10 - 5 = 5 of them) and those that are good for lunch but not breakfast (8 -5 = 3 of them) must be good for dessert precisely because they are not good for both breakfast and lunch. Clara Blackstone's diagram is correct if you just look at the blue numbers where there are two.

Michael Boyd - 2 years, 1 month ago

A probability problem by Clara Blackstone

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