The probabilities that three men hit a target are, respectively \[0.3\], \[0.5\] and \[0.4\]. Each fires once at the target. (As usual assume that the three events that each hits the target are independent).
a)Find the probability that they all : (i) hit the target
(ii) miss the target

Hint: Here we need to find the probability that all three men will hit the target and all three miss the target. Since these three events are independent events. The probability that they will hit the target will be equal to the product of probabilities of hitting the target by each man. We will find the probability that they miss the target, for that we will first probability of each man of missing target and we will find their product to obtain the required probability.

Complete step-by-step answer:
It is given that,
Probability of hitting the target by first man \[ = 0.3\]
Probability of hitting the target by second man \[ = 0.5\]
Probability of hitting the target by third man \[ = 0.4\]
It is also given that the three events that each hits the target are independent.
Now, we will find the probability of hitting the target by all three men.
Probability of hitting the target by all three men \[ = \] product of probability of hitting target by each man
Therefore,
Probability of hitting the target by all three men\[ = 0.3 \times 0.4 \times 0.5 = 0.06\]
Now, we will find the probability of missing the target by all three.
Probability of missing the target by first man \[ = 1 - 0.3 = 0.7\]
Probability of missing the target by second man \[ = 1 - 0.5 = 0.5\]
Probability of missing the target by third man \[ = 1 - 0.4 = 0.6\]
It is also given that the three events that each hits the target are independent.
Now, we will find the probability of missing the target by all three men.
Probability of hitting the target by all three men\[ = \] product of probability of missing target by all three men
Therefore,
Probability of missing the target by all three men\[ = 0.7 \times 0.6 \times 0.5 = 0.21\]
Hence, the probability that they all hit the target is 0.06 and the probability that they all miss the target is 0.21.

Note: We have calculated the value of probability of hitting and missing the target by all three men. A probability is defined as the chances of occurrence of any random event. Probability is equal to the ratio of the favorable outcomes to the total outcomes. We need to always remember that the sum of probability of all the events in a sample space is equal to one.