Bill Gates Investment Problem

Question

Bill Gates will invest in only one of 5 companies: $C_1$, $C_2$, $C_3$, $C_4$ or $C_5$. I will make a lot of money if I invest in the same company. I decide to invest in company $C_1$ and I inform Bill Gates. He tells me that he is going to invest in neither company $C_5$ nor $C_4$.

Should I invest in a company other than $C_1$?

I believe that there won't be any difference. 1/3 chances for each of the remaining companies. My friend told me that the correct solution would be to change companies. What does the math community think?

Answer 1

Your friend is absolutely right. When you chose your company, your odds were $20%$ as you had to pick $1$ out of $5$ possibilities. However, after he eliminates $2$ choices, there are only $3$ possibilities so if you switch, you will have an odds of $100%-20%=80$ chance for the companies $2$ and $3$ since Bill Gates has told you that he has not invested in the other companies. Now divide by $2$ to get chance for each company, which is $20$ percent. Think about it: the odds of you getting it right without any information is lower than using the information provided. Information changes probability.

This is a variation of the famous "Monty Hall" problem. I suggest that you read up on it here https://en.wikipedia.org/w/index.php?title=Special:Search&search=Monty+Hall+problem

or watch a youtube video on it. There are thousands of sources discussing the problem in detail.

Answer 2

I am assuming that Bill Gates chose $C_4$ and $C_5$ to reveal by randomly selecting two companies from $C_2,C_3,C_4,C_5$ which he did not invest in. This is the assumption which makes the problem interesting.

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Imagine repeating this experiment many times in a row. Each time, Bill Gates invests in a random company, you select company $1$, then Bill Gates reveals two companies that he does not invest in (not always $C_4$ and $C_5$). Suppose you never switch companies throughout this repeated experiment; how often would you win on average?

A little thought shows you would win $20\%$ of the time, because $20\%$ of the time, Bill Gates invests in company $1$, and that is the only way you win. Bill Gates revealing the two companies he did not invest in does not affect the outcome at all, since you decided not to switch.

Therefore, the probability your original guess is correct is $20\%$. By symmetry, the other two companies have a $40\%$ chance of being correct. Therefore, you should switch companies.

Here is another way to think about it. There is an $80\%$ chance you initially picked the wrong company. In that case, one of $C_2$ and $C_3$ is the correct company, each equally likely. Therefore, the probability of winning if you switch companies is $80\%\times 50\%=40\%$.