Example There are 25 sets of plastic models, of which 2 sets have missing parts. When the customer arbitrarily chooses three sets, find the probability that they are all complete sets

Answer Trials that satisfy the following 1 to 3 are called bf Bernoulli trials.

1. In each trial, the only question is whether or not the event occurs.
2. Each trial is statistically independent.
3. The probability that the target event will occur is constant throughout each trial.

Let $p$ be the probability that a certain event $X$ will occur in one trial. In the n-time Bernoulli trial sequence, the probability that the event $X$ will occur exactly $i$ times is

$$P(X = i) = \binom{n}{i}p^{i}(1-p)^{n-i}$$

At this time, the probability distribution of $X$ is called the binomial distribution and is expressed as $X \sim B(n,p)$.

Those that satisfy the following conditions 1 to 5 are called Poisson process.

1. Events can occur randomly at any time.
2. The occurrence of an event in a given time interval is independent of other intervals that do not overlap.
3. The probability of occurrence of an event in a minute time $\Delta t$ decreases in proportion to $\Delta t$.
4. The probability that an event will occur more than once during a minute time $\Delta t$ is negligible.
5. The average number of occurrences of the event $\lambda$ during the time $t$ is about 5 or less.

$Let$X be the number of events that occur in the Poisson process

$$P_{r}(X = r) = \frac{\lambda^{r}}{r!}e^{-\lambda}$$

and we write $X \sim P_{o}(\lambda)$. Here, $\lambda$ is the average number of events in the Poisson process.

The Poisson process includes scratches on the tape, incoming calls to the switchboard, and broken light bulbs.

Answer to the problem

The trial to take out the plastic model is Bernoulli trial. Let $X$ be the number of sets of plastic models that the part is missing，Then $X \sim B(3，\frac{2}{25})$．Thus, The probability that all three selected sets are complete sets is

$$P(X = 0) = \binom{3}{0}(\frac{2}{25})^{0}(1 - \frac{2}{25})^{3} = 0.7789$$