An automated programmable machine is independently attempting to hit
specific targets, A, B, C across 10 attempts. The probability of successfully hitting one of
the targets is 0.9. Otherwise, it fails and misses with probability 0.1. It is known that it hits
only A or C with probability 0.4 and only B with probability 0.6.
a. (4 marks) Define the distribution for X for hitting a target and calculate the probability
that any 6 out of the 10 attempts manage to successfully hit any target.
b. (5 marks) What is the probability that any 6 out of the 10 attempts manage to hit only on
the A or C targets? The other attempts could have either hit the B target or missed.
c. (4 marks) Use Venn diagrams and probability axioms to prove that
(?! n ?” n ?#
$ ) n (?! n ?”
$ n ?#) = Ø
where ?!, ?”, ?# are three sets.
d. (4 marks) Let one of the three attempts be tracked with a red indicator. The company
would like to analyse the scenario observed that usually one of these attempts fail to hit
when the red indicator is present. What is the probability that the red indicator hits a
target and one of the other two attempts fail? Hint: Use a result similar to part c.
e. (5 marks) If two consecutive attempts manage to hit target B then what is the probability
the next attempt hits target B out of ten attempts?