There is a sports gun whose probability of working is $0.8$, i.e. it functions $80\%$ of the time I operate it. The probability that it hits the target accurately when i use it is $0.6$, i.e. $60\%$ of the time the gun hits the target accurately. 0.8 is the serviceability of the gun, the gun is serviceable for firing. 0.6 is the probability that i as a shooter hit the target accurately. i can hit target accurately only 6 out of 10 times.
- What is the probability that the gun when fired hits the target, on any given day if I use the gun ?
- I understood that functioning of gun and target accurately being hit are dependent events. Because if the gun is not functioning, I can’t hit the target. Also I can’t change the sequence of events. Due to these reasons I assumed they are dependent events, is my assumption correct? Are they dependent events?
- Is the data adequate to calculate probability of hitting the target if I pick up the gun any day?
Yes, the event "The gun shoots and hits the target" and "The gun is working, i.e. it shoots" are dependent. We can actually prove that this is the case if we can show that $$\mathbb{P}(\text{ The gun shoots and hits the target}) \neq \mathbb{P}(\text{ The gun shoots and hits the target} \ | \ \text{ The gun is working }).$$ In fact we know that the second probability is $0.6$ by the assumptions (as clarified in the comments). The first probability can be rewritten as $$\mathbb{P}(\text{ The gun shoots and hits the target} \ | \ \text{ The gun is working }) \mathbb{P}(\text{ The gun is working }),$$ since on the envent "The gun is not working" the probability of hitting the target is zero. You know both terms in this product, so a multiplication will answer everything.