What is the probability that the bonus scheme will be introduced?

Calculation:

Given,

The probability that A , B , and C  become managers are:

\( P(A) = \frac{3}{10}, \, P(B) = \frac{1}{2}, \, P(C) = \frac{4}{5} \)

The conditional probabilities that the bonus scheme is introduced are:

\( P(D|A) = \frac{4}{9}, \, P(D|B) = \frac{2}{9}, \, P(D|C) = \frac{1}{3} \)

The total probability that the bonus scheme will be introduced is given by:

\( P(D) = P(D|A)P(A) + P(D|B)P(B) + P(D|C)P(C) \)

Substituting the values:

\( P(D) = \left(\frac{4}{9} \times \frac{3}{10}\right) + \left(\frac{2}{9} \times \frac{1}{2}\right) + \left(\frac{1}{3} \times \frac{4}{5}\right) \)

Now, simplifying the terms:

\( P(D) = \frac{12}{90} + \frac{2}{18} + \frac{4}{15} \)

The least common denominator (LCD) is 90. So:

\( P(D) = \frac{12}{90} + \frac{10}{90} + \frac{24}{90} \)

Adding the fractions gives:

\( P(D) = \frac{46}{90} = \frac{23}{45} \)

Hence, the correct answer is Option 3.