In this problem:

• $A$ denotes the event of placing a straight bet on a lottery and winning.
• There are 7,200 different possible outcomes (four-digit numbers with repetition allowed).
• Only one outcome corresponds to a win.

To calculate the probabilities:

1. Probability of winning, $P(A)$: The probability of winning is the number of favorable outcomes (1 winning number) divided by the total number of possible outcomes (7,200): $P(A) = \frac{1}{7200}$ So, $P(A) = 0.00013889$ or about 0.01389%.

2. Probability of losing, $P(\overline{A})$: The complement of event $A$ (denoted $\overline{A}$) represents the event of not winning the lottery, which is losing. The probability of this event is: $P(\overline{A}) = 1 - P(A)$ Substituting the value of $P(A)$: $P(\overline{A}) = 1 - \frac{1}{7200} = \frac{7199}{7200} = 0.9998611$ or about 99.986%.

Summary:

• $P(A) = 0.00013889$ (or 0.01389%)
• $P(\overline{A}) = 0.9998611$ (or 99.986%)

Do you want more details or have any questions?

Here are some related questions:

1. How would the probabilities change if the lottery had 10,000 possible outcomes?
2. How do probabilities of complementary events relate in general?
3. What happens to the probability if you play multiple independent lottery games?
4. Can the probability of a lottery win be increased through any strategies?
5. How are probabilities used in other types of gambling games, like poker?

Tip: In lotteries, the odds are typically very low due to the high number of possible outcomes, and complementary probabilities (not winning) are close to 1.