Delta Inc $($DI$),$ Elmo Trust $($ET$)$ and Frank Partnerships$($FP$)$ handle legal cases involving the breach of personal data under the Personal Data Protection Act $($PDPA$)$ on behalf of the Personal Data Protection Commission $($PDPC$).$ ET takes $20\%$ of all suits in Singapore, FP handles $40\%$ and DI takes the other $40\%$ of cases. ET wins $70\%$ of its cases for the suits by PDPC against businesses suspected of breach. FP wins $60\%$ of its cases whilst DI wins only $40\%$ of its cases.

a$.$ Develop a probability tree showing all marginal, conditional, and joint probabilities.

b$.$ Develop a joint probability table.

c$.$ Using Bayes$$ rule, determine the probability that the FP firm handled a particular case, given that the case was won.