Difficult counting problem

Question

A bookie offers two bets I and II on a particular football match: I You win this bet if the total number of goals scored in the first half of the game is at most 1. II You win this bet if the total number of goals scored in the entire game is at most 2. (a) If goals are scored according to a Poisson process with mean λ per game, are you more likely to win bet I or bet II? (Your answer will depend on λ.) (b) For what value of λ are the probabilities of winning the two bets the same?

Answer 1

1. If the expected total goals per game are less than ~1.23, Bet II is more likely to win.
2. If the expected goals are more than ~1.23, Bet I is the better choice.
3. And at exactly:

λ≈1.23λ≈1.23​…the two bets are equally likely.