- Winning and losing are complementary events, so their probabilities must sum to 1.
- Subtract the win probability from 1: \(1 - \frac{2}{9} = \frac{7}{9}\).
- The probability that Andrew loses is \(\boxed{\frac{7}{9}}\).
Explanation
- Analyze the relationship between winning and losing probabilities
The probability of winning and the probability of losing a tennis match are complementary events, meaning they are the only two possible outcomes, and their sum must equal \(1\). Given that the probability of Andrew winning is \(P(\text{Win}) = \frac{2}{9}\), we can determine the probability of him losing using the complement rule.
- Calculate the probability of losing
To find the probability that Andrew will lose, we use the rule for complementary events:
\[P(\text{Lose}) = 1 - P(\text{Win})\]
Substituting the known value of \(P(\text{Win}) = \frac{2}{9}\) into the equation, we get:
\[P(\text{Lose}) = 1 - \frac{2}{9}\]
- Final calculation and conclusion
To subtract the fraction from 1, we rewrite 1 as \(\frac{9}{9}\):
\[P(\text{Lose}) = \frac{9}{9} - \frac{2}{9} = \frac{9 - 2}{9} = \frac{7}{9}\]
Thus, the probability that Andrew will lose the match is \(\frac{7}{9}\).
Examples
This concept of complementary events is vital in real-world scenarios where you have a set of possible outcomes. For instance, if you know the probability of your train being on time is \(0.85\), you can quickly calculate the probability of it being late or delayed as \(1 - 0.85 = 0.15\). This helps in planning travel more effectively by understanding the likelihood of unexpected delays.