Please help me understand the problem

Question

Sheldon Ross's A First Course in Probability, Chapter 3, Problem 3.76: Consider an unending sequence of independent trials, where each trial is equally likely to result in any of the outcomes 1, 2, or 3. Given that outcome 3 is the last of the three outcomes to occur, find the conditional probability that (a)the first trial results is outcome 1; (b)the first two trials both result in outcome 1.

I can't understand what does it mean,I think there are no influence about the first trial results and the last three outcomes,because it says they are independent trials.

Answer 1

I had trouble understanding it too, but looking up the solution resolved the ambiguity. It means that in the results from the unending sequence of trials both of the outcomes $1$ and $2$ occur before the outcome $3$ does. This implies that neither of the first two trials can result in outcome $3$.

Answer 2

I explain the first part $(a)$. You have to calculate a conditional probability as $P(A|B)=\frac{P(A\cap B)}{P(B)}$.

The given event is $B$. This is where you have the sequence $\color{green}x\color{blue}{xx...xxx}\color{red}3 \ \ (\textrm{seq. I})$, where $x$ is a $1$ or a $2$. The event $A$ occurs, if the first trial results in 1. Thus the event $A\cap B$ is the sequence $\color{green}1\color{blue}{xx...xxx}\color{red}3 \ \ (\textrm{seq. II})$.

So you need to calculate the probability of getting sequence $\textrm{II}$ when sequence $\textrm{I}$ has already been given.

Answer 3

Let random variables $X,Y,Z$ denote the first trial in which these respective outcomes occur.

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The condition is that the first occurrence of outcome 3 is after the first occurrence of the other two outcomes.

$$\{X<Z\} \cap \{Y<Z\}~=~\{X<Y<Z\}\cup\{Y<X<Z\}$$

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The event of the first problem is that the outcome 1 first occurs in the first trial: $\{X=1\}$. So you must evaluate the conditional probability:

$$P_1:=\mathsf P(X=1\mid X<Z, Y<Z)$$

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The event of the second problem is that the outcome 1 first occurs in the first trial, while both other outcomes first occur after trial 2: $\{X=1\}\cap \{2<Y\}\cap \{2<Z\}$ . So you must evaluate the conditional probability:

$$P_2:=\mathsf P(X=1, 2<Y, 2<Z\mid X<Z, Y<Z)$$

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Use Bayes' Rule and arguments about symmetry to evaluate these conditional probabiltities