Certainly! Let's go through the questions step-by-step.
### Question 1:
If you play the game 30 times, how many times would you expect to win?
To find the expected number of wins, we'll use the concept of expectation in probability. The expectation is given by multiplying the probability of a single event by the number of trials.
Here, the probability of winning a game is 4.8%, or 0.048. If you play the game 30 times, the expected number of wins is:
[tex]\[ \text{Expected wins} = \text{Probability of winning} \times \text{Number of games played} \][/tex]
[tex]\[ \text{Expected wins} = 0.048 \times 30 \][/tex]
[tex]\[ \text{Expected wins} = 1.44 \][/tex]
Therefore, if you play the game 30 times, you would expect to win approximately 1.44 times.
### Question 2:
After how many times playing the game could you reasonably expect to win?
To determine the number of trials after which you could reasonably expect to win at least once, we'll use the reciprocal of the probability. This approach is based on the idea that, on average, it takes a certain number of trials to achieve one successful outcome.
Given the probability of winning a single game is 0.048, the reasonable number of trials to expect at least one win is:
[tex]\[ \text{Reasonable play count} = \frac{1}{\text{Probability of winning}} \][/tex]
[tex]\[ \text{Reasonable play count} = \frac{1}{0.048} \][/tex]
[tex]\[ \text{Reasonable play count} \approx 20.83 \][/tex]
So, you could reasonably expect to win at least once after playing the game approximately 20.83 times.
### Final Answers:
1. If you play the game 30 times, you would expect to win approximately 1.44 times.
2. You could reasonably expect to win after playing the game approximately 20.83 times.
