An automated programmable machine is independently attempting to hit specific targets, A, B, C…

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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?

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