Binomial distribution

Exercise 8

1. For the binomial distribution $B(8,0.4), \ B(8,0.2)$, find the probability disribution.

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

3. It is said that $40\%$ will die within a certain period of time when a certain reagent is injected into a mouse. When the reagent is injected into 8 mice, answer the following questions.

(a)

Find the probability that all 8 mice will survive for a certain period of time or longe．

(b)

How many out of eight can survive with a probability of $95\%$?

Answer

1. $p_{r} = P(X = r) = \binom{8}{r}(0.4)^{r}(0.6)^{1-r}$. Then

$$\begin{array}{\vert c\vert c\vert c\vert c\vert c\vert c\vert... ....2322 & 0.1239 & 0.0413 & 0.0079 & 0.0007 \\ \hline \end{array}$$

$p_{r} = P(X = r) = \binom{8}{r}(0.2)^{r}(0.8)^{1-r}$. Then

$$\begin{array}{\vert c\vert c\vert c\vert c\vert c\vert c\vert... ...2 & 0.0011 & 0.0001 & 2.56 \times 10^{-6} \\ \hline \end{array}$$

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

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

3. Let $X$ be the number of mice that cannot survive for a certain period of time after injection and die. Then $X \sim B(8,0.4)$ andwe can use the probability distribution obtained in I．

(a) The probability that all eight will survive for a certain period of time is

$$P(X = 0) = \binom{8}{0}(0.4)^{0}(0.6)^{8} = 0.0168$$

(b) Assuming that $r$ is the number of mice that cannot survive with a probability of $95\%$ or more, the distribution function $F(x)$ of $X$ satisfies

$$F(r) = P(X \leq r) = \sum_{k \leq r} r\binom{8}{k}(0.4)^{k}(0.6)^{8-k} \geq 0.95$$

Using the probability distribution of I, we have $P(X = 0) = 0.168, P(X=1) = 0.0896$. Then $r = 1$ and the number of mice that can survive for $95\%$ is 7 or more