(a) From the table, \(\sum f = x + 155\)
P(obtaining a 3) = \(\frac{\text{no of times a 3 was obtained}}{\sum f}\)
\(\frac{x}{x + 155} = 0.225\)
\(x = 0.255x + (155 \times 0.225)\)
\(x - 0.225x = 34.875 \)
\(0.775x = 34.875\)
\(x = \frac{39.875}{0.775}\)
\(x = 45\)
\(\sum f = 155 + x = 155 + 45 = 200\)
Hence, the die was thrown 200 times.
(b) Set of even numbers = {2, 4, 6}
Set of prime numbers = {2, 4, 5}
Therefore, \(N \cup M = (2, 3, 4, 5, 6)\)
\(n(N \cup M) = 30 + 45 + 28 + 40 + 32 = 175\)
\(P(N \cup M) = \frac{175}{200} = 0.875\)