## Betcha Find a Chocolate

Laura saw that there were three pieces of candy left in a bowl on the kitchen table. She knew they were either chocolates or caramels but didn’t know how many there were of each. What is the probability that there is a piece of chocolate candy in the bowl?

7/8.

The possible combinations are:

1. Chocolate Chocolate Chocolate

2. Caramel Caramel Caramel

3. Chocolate Chocolate Caramel

4. Caramel Caramel Chocolate

5. Chocolate Caramel Caramel

6. Caramel Chocolate Chocolate

7. Chocolate Caramel Chocolate

8. Caramel Chocolate Caramel

Number 2 is the only one that doesn’t have chocolate, thus Laura’s love of chocolate will more than likely be satiated.

4 Comments on "Betcha Find a Chocolate"Mark says

May 4, 2019 @ 09:26

This answer is not correct. You’ve double counted possible combinations. For instance: Caramel Caramel Chocolate and Chocolate Caramel Caramel are the same combination, just different order which does not matter in this problem. There are really only 4 combinations with only 1 not having chocolate. Therefore the answer is 75%.

Dan says

May 4, 2019 @ 11:11

The order does matter because you have to account for all possible outcomes. The two you described are both separate, possible outcomes.

AlchemiCailleach says

June 10, 2019 @ 20:57

She does not know how many of each were in the bowl, but given that the candies were either chocolates or caramels, we will assume for now at least, that the likelihood for each candy is 50%.

There IS an error in the combinations for the answer.

Using binary, we can see what all the combinations are supposed to be.

Notably, across 8 possible combinations (drawing 24 total candies), there end up being 12 chocolate and 12 caramels still – consistent with the 50% probability of each candy being either chocolate or caramel.

1: 000 c – c – c 0 choc

2: 001 c – c – ch 1 choc

3: 010 c – ch – c 1 choc

4: 011 c – ch – ch 2 choc

5: 100 ch – c – c 1 choc

6: 101 ch – c – ch 2 choc

7: 110 ch – ch – c 2 choc

8: 111 ch – ch – ch 3 choc

ThomasSchmitt says

July 8, 2019 @ 19:43

Dan, I agree with you and the original answer. All possible outcomes must be included in the answer.

Mark: Your double counting comment has intuitive appeal—a chocolate piece tastes the same whether it’s selected first, second or third, but as Dan explains, “The two (outcomes) you described are both separate, possible outcomes.”

AlchemiCailleach, I agree with your binary classification of possible outcomes, but how is it any different than the original answer? A successful outcome for Laura (i.e., a “1” in your binary classification) is getting at least one chocolate. It doesn’t matter whether she chooses 1, 2 or 3 chocolates. Therefore, outcomes 001, 010, 011. 100, 101, 110 and 111 are all successes, and this takes us back to the original answer, 7/8. Your answer (based on a 50% assumption) is only valid for the very first selection from the bowl.

What if a 24-piece bowl originally contained 6 chocolates and 18 caramels? Your 50% answer wouldn’t hold up at all because the individual selection probability would be changed to 25%. Laura’s selection outcomes would clearly be conditional on the changing probability of each successive selection leading up to her choices. For example, the person who chooses first would have a 25% (6/24) chance of selecting a chocolate piece. If the first selection was chocolate, the next selection would have a 21.7% (5/23) probability of being chocolate. If the first selection was caramel, the next selection would have a 26% (6/23) probability of being chocolate. And so on.

But based on the wording of the original question, we cannot assume an equal number of caramel and chocolate pieces to begin with. This brainteaser is flawed and unsolvable without this information.

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