A die is thrown two times and the sum of the scores...

A die is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of 4 . Then the conditional probability that the score 4 has appeared atleast once is :
[Main Sep. 03, 2020 (I)](a) $\frac{1}{4}$
(b) $\frac{1}{3}$
(c) $\frac{1}{8}$
(d) $\frac{1}{9}$

Key Concepts

Conditional Probability

Conditional probability involves finding the probability of an event occurring given that another event has already occurred. It requires a recalculation of the sample space based on the new condition and focuses only on outcomes that satisfy the given event, thereby altering the original probability computation.

Sample Space

Understanding the sample space means recognizing the complete collection of all possible outcomes of an experiment. In probability problems, especially those involving random processes like dice rolls, accurately defining the sample space is essential for calculating event probabilities correctly.

Counting Techniques

Counting techniques are fundamental in probability for determining the number of favorable outcomes versus the total number of outcomes in a sample space. They include methods like listing outcomes, using basic combinatorial principles, and applying rules of chance to compute probabilities in complex scenarios.

Complementary Events and 'At Least Once' Scenarios

Calculating the probability of an event occurring at least once typically involves understanding and applying the complementary event principle. Instead of directly computing the probability of one or more occurrences, it can be easier to calculate the probability of the event not occurring at all and subtract this from one to obtain the desired probability.

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