An event has odds in favour $4 : 5$. Then,the probability that the event occurs is:

The odds against an event are $5 : 2$ and the odds in favor of another independent event are $6 : 5$. What is the probability that at least one of the events occurs?

If the odds against solving a question by three students are $2:1$,$5:2$,and $5:3$ respectively,then the probability that the question is solved by only one student is

Let $A$ and $B$ be independent events with $P(A)=0.3$ and $P(B)=0.4$. Find $P(A \cup B)$.

If $A, B, C$ are three events associated with a random experiment,prove that $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$.

In a certain population,$10\%$ of the people are rich,$5\%$ are famous,and $3\%$ are both rich and famous. The probability that a person picked at random from the population is either famous or rich but not both,is equal to