Consider an unending sequence of independent trials, where...

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 in outcome 1 ; (b) the first two trials both result in outcome $1 .$

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 in outcome 1 ;
(b) the first two trials both result in outcome $1 .$

Key Concepts

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. In this problem, the calculation of the probability that the first (or the first two) trial(s) result in a specific outcome is performed under the condition that outcome 3 turns out to be the last among the three possible outcomes to appear. This conditioning changes the sample space and often requires careful reasoning to adjust for the given scenario.

Order of First Occurrences and Permutation Symmetry

This concept involves analyzing the order in which outcomes appear for the first time in a sequence of trials, and it makes use of the symmetry inherent in equally likely outcomes. When it is known, for instance, that one outcome (here outcome 3) is the last to appear among a set, the chance of another outcome appearing at a particular position (such as first) can be reasoned through by considering the equally likely arrangements (or permutations) of the order in which the outcomes occur. This symmetry simplifies the analysis by reducing the problem to counting the favorable orders over all orders that satisfy the given condition.

Independent Trials

This concept refers to the idea that the outcome of any trial in a sequence does not affect the outcomes of other trials. In the context of the problem, each trial is conducted under the same probability conditions, and every trial's outcome is independent of the previous ones. This property simplifies calculations by allowing the multiplication of probabilities of individual events when considering joint occurrences.

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