Final answer:
Considering the expected outcomes, it is improbable that playing the casino game would be more profitable than working hard all summer since the expected loss of playing the game is $90, compared to the guaranteed $450 from working.
Step-by-step explanation:
To determine whether playing the casino game yields more money than hardworking all summer, we need to compare the expected winnings from the casino to the guaranteed earnings from working. The colleague's earnings are $5 per day for 90 days, totaling $450. The expectation of winning in the casino per game can be calculated using the provided probabilities: E(X) = (0.1 × $50) + (0.3 × $10) - (0.6 × $10) = $5 - $6 = -$1. Thus, on average, you'd expect to lose $1 per game in the long term. Playing 90 games, your expected total loss would be 90 × -$1 = -$90.
Since your colleague is guaranteed to make $450 and the expected loss in the casino is $90, it is improbable that playing the casino game would yield more money than working hard all summer. Therefore, the answer is that the probability of the casino game being more profitable is technically zero because the expectation is a net loss.