Total probability theorem visualisation I recently started doing this topic but I am having trouble identifying the sample sample space and solving it .I will give u an example in case Eg-When a missile is fired from a ship,the probability that it is intercepted is 1/3.The probability that the missile hits the targer ,given it is not intercepted is 3/4.If three missile are fired indpendently from ahip,the probability that all three hits the target. My analysis-Basically calculating probability of the missile hitting one target and multiplying it to itself three times will give me the probability.But I can't visualise the sample space and also the events that are mutually exclusive and exhaustive.any help

A suitable sample space would be the results of the three missile launches: intercept, miss, or hit. $\\{{\sf I, M, H}\\}^3$ is an acceptable representation for this. However, you hardly need to use this.

The results for a particular missile launch are _mutually exclusive and exhaustive_ events, since a launched missile may either be exactly one of intercepted, miss, or hit; not anything else, and not any combination.

The three launch results are pairwise independent events, since the results for any two particular missiles are presented as having no influence on each other.

Your favoured event is $\\{({\sf H,H,H})\\}$ and indeed is $\mathbb P\\{({\sf H,H,H})\\}= {(\tfrac 23\tfrac 34)}^3$, which comes from the probabilities given and the pairwise independence between the results for different missiles. (Also the definition of conditional probability).