Observation of events and analysis of the associated Hypotheses

[Moara]

Homework Statement

A scientist observs the occurence of an event A as a result of some experiment. He believes that the only possible explanations for the occurence of event A are three different hypothesis, $H_1, H_2, H_3$.

With hypothesis $H_1$, the experiment produces A in $10\%$ of time, when repeted indefinitely. With $H_2$, A is observed $1\%$ of time and, under $H_3$, A is observed in $39\%$ of time.

The scientis decides that $H_3$ is the most likely explanation and that the probability that $H_3$ is true is: $\frac{0.39}{0.1 + 0.01 + 0.39} = 0.78$

a) What considerations are being assumed as true by the scientist?

b) The probability $0.78$ admits the interpretation of relative frequencies ? Justify

c) Suppose that the experiment consists in a lab test made with a blood sample from a person randonly choosen of a population. The hypothesis $H_i$ is that the individual's blood is of type i. It is known that $50\%$ of the population has blood type $1$. $45\%$ has blood type $2$ and the remaining part has blood type $3$. In this conditions, find which of the hypothesis is mos likely, given that the event A was observed.

Relevant Equations

$P =\frac{ n(favorable)}{N}$, $P(A|B) = \frac{P(A \ and \ B)}{P(B)}$

For letter a), i think that he is assuming that each hypothesis is independent, and that they are mutually exclusive.For letter b), I understand that it indeed admits the relative frequency interpretation, since the the experiment is being produced several times.

For letter c) we do the conditional probability, $P(i | A) = \frac{P(i \ and \ A)}{P(A)}$, $P(A) = 0.1 \cdot 0.5 + 0.01 \cdot 0.45 + 0.39 \cdot 0.05$ (which does not depends on $i$), and $P(i \ and \ A)$ is greater for $i = 1$, $P(1 \ and \ A) = 0.1 \cdot 0.5$, So the most likely hypothesis is $H_1$

 

[FactChecker]

The information that A occurred can be used to update the probabilities of a random variable. The calculation used in (a) and (b) assumes that the prior probabilities of $H_1, H_2, H_3$ being true are all equal.

 

[Moara]

The information that A occurred can be used to update the probabilities of a random variable. The calculation shown assumes that the prior probabilities of $H_1, H_2, H_3$ being true are all equal.

you mean my calculations in letter c) ?

 

[FactChecker]

you mean my calculations in letter c) ?

No, sorry. I meant the calculations referred to in (a) and (b).

 

[PeroK]

you mean my calculations in letter c) ?

When you did the calculation in part c) you used the fact that the different hypotheses were not initially equally likely.

The scientist's simpler calculations, however assumed that the hypotheses had equal prior probabilities.

I agree that there is a further assumption that the hypotheses are mutually exclusive. But, they are clearly not independent.

 

[Moara]

When you did the calculation in part c) you used the fact that the different hypotheses were not initially equally likely.

The scientist's simpler calculations, however assumed that the hypotheses had equal prior probabilities.

I agree that there is a further assumption that the hypotheses are mutually exclusive. But, they are clearly not independent.

so you agree with my calculations in c), but don't agree with the scientist's assumption that the events are independent? And letter b) could you clarify what that relative frequency interpretation mean ?

 

[Moara]

No, sorry. I meant the calculations referred to in (a) and (b).

So you agree with my part c) calculation ? What about letter b), as told before, I don't think I truly understand what that relative frequency question mean.

 

[FactChecker]

So you agree with my part c) calculation ?

Yes, I agree with your part (c).

What about letter b), as told before, I don't think I truly understand what that relative frequency question mean.

Could that be "omits the interpretation of relative frequencies"? It does omit consideration of the relative frequencies of the three hypotheses. I also do not understand what "admits the interpretation of relative frequencies" means.

 

[PeroK]

so you agree with my calculations in c), but don't agree with the scientist's assumption that the events are independent? And letter b) could you clarify what that relative frequency interpretation mean ?

I agree with your calculations in c). I don't agree with your idea that the hypotheses are independent. Mutually exclusive implies dependent.