Answer:
a)
Operating room x    1         2      3       4
P(x)                          0.15  0.25  0.4   0.2
b)
The graph of probability distribution is in attached image. 
c)
All the probabilities lies in the range (0 to 1) and the probabilities add up to 1 so, the computed probability distribution is the valid probability distribution.
Step-by-step explanation:
Number of operating rooms   Days
1                                                  3 
2                                                 5
3                                                 8
4                                                 4
a)
Sum of frequencies=1+2+3+4=10
Operating rooms x   frequency f  Relative frequency
1                                    3                    3/20=0.15
2                                   5                   5/20=0.25
3                                   8                   8/20=0.4
4                                   4                   4/20=0.2
So, using the relative frequency approach, a discrete probability distribution for number of operating rooms in use on any given day is 
Operating room x    1         2      3       4
P(x)                          0.15  0.25  0.4   0.2
b)
The graph of probability distribution is in attached image. 
c)
There are two conditions for a probability distribution to be valid
1. All probability must ranges from 0 to 1.
2. All probabilities must add up to 1.
We can see that all the probabilities lies in the range (0 to 1), so, condition 1 is satisfied.
For condition 2,
sum[p(x)]= 0.15+0.25+0.4+0.2=0.4+0.6=1.
As the probabilities add up to 1, so the condition 2 is also satisfied.
Thus, the computed probability distribution is the valid probability distribution.