Probability of correctly guessing some factor from a sample with 3 factors

Question

Suppose I have a situation where I have a sample with 3 factors: {cats, dogs, hamsters}

We have 105 cats, 62 dogs, and 12 hamsters.

Now, the probability of drawing a cat is simple: $\frac{105}{(105+62+12)}$

But now suppose there is a curtain behind a randomly selected animal in my sample. I need to guess what the animal is. If I follow a decision rule where I guess the animal species based on the probability distribution, can I determine the probability of error?

So my decision for guessing is to guess cat 58.66% of the time, dog 34.64% of the time, and hamster 6.7% of the time.

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E.g., I guess cat for $\frac{105}{(105+62+12)}\times100\%=58.66\% $ cases then is my probability of being correct $.5866^2 \times 100\%= 34.41\%?$

Answer 1

If your guess is decided probabilistically (i.e. your guess is determined from a distribution rather than what you think or want your guess to be) then the short answer is: yes.

Based on this information, your guess and the animal behind the curtain may share the same distribution pattern, but they are actually independent of one another.

So for your example, what you are finding is the probability of the event where a cat being behind the curtain AND your decision rule "rolls" on you deciding to guess cat on this turn. That probability is, as you found, $0.3441$.

The probability that your decision rule is correct in general is the sum of these three cases: Guessing Cat and actually having a cat behind the curtain, guessing dog and actually having a dog, and guessing hamster and actually having hamster.

Answer 2

Let $p_C,p _H, p_D$ be the probability of drawing a cat/hamster/dog.

If you follow the deterministic strategy "guess the most probable one" (which is the optimal strategy, if you want to minimize the probability of error), then you would always guess "cat", and the probability of error is $p_D(e) = 1 - p_C = p_H + p_D$

If you instead guess randomly, guessing "cat" with probability $p_C$ and so on, then the probability of getting it right is $p_C^2+p_H^2 +p_D^2$, and the probability of error is $p_R(e) = 1- (p_C^2+p_H^2 +p_D^2)$