The minimum number of test cases needed to cover every unique sequence of up to 3 states/2 transitions starting in the "Start" state and ending in the "End" state is 4. This is because there are 4 unique sequences of up to 3 states/2 transitions starting in the "Start" state and ending in the "End" state:
Start -> Bet -> End
Start -> Deal -> End
Start -> 1st Deal -> End
Start -> 2nd Deal -> End Reference: ISTQB Certified Tester Foundation Level (CTFL) v4.0 sources and documents.