2
$\begingroup$

Suppose you throw 1000 darts where each dart has 0.5 probability of scoring. For the first 500 darts each is worth 1 point, for the second 500 darts each is worth 3 points. If you score 1500 points, how many 3 point darts have you scored most likely?

If I use the likelihood method, I need to maximize the following function: $$ \binom{500}{k} \binom{500}{1500-3k}$$ where $k$ ranges from $334$ to $500$. The maximum happens at $k = 398$.

However, if I try to find the expectation I get a different value. From the question we can model the first dart as a binomial distribution with $B_1(500,0.5)$ and the second dart as a binomial distribution $B_2$ with same parameters. Then we need to find $$ E [B_2 | B_1 + 3B_2 = 1500]$$ which turns out to be $375$.

My question is which approach should I choose given that I obtain different values for each?

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ "Most likely" has to be the mode, since it has the highest probability mass. $\endgroup$
    – TenaliRaman
    May 21, 2014 at 13:26
  • 2
    $\begingroup$ +1 to @TenaliRaman's comment. This is an example where the mode of the probability mass function is different from its integral, which is the expectation. Which one you need really depends on what you will do with the result. $\endgroup$
    – Stephan Kolassa
    May 21, 2014 at 13:36

1 Answer 1

Reset to default
0
$\begingroup$

Partially answered in comments:

"Most likely" has to be the mode, since it has the highest probability mass. – TenaliRaman

+1 to @TenaliRaman's comment. This is an example where the mode of the probability mass function is different from its integral, which is the expectation. Which one you need really depends on what you will do with the result. – Stephan Kolassa

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.