Please provide a clear and step-by-step solution for a scientific problem in the categories of Chemistry, Physics, or Mathematics. The problem will specify the unit of measurement, which should not be included in the answer. Express the final answer as a decimal number with three digits after the decimal point. Conclude the answer by stating 'Therefore, the answer is \boxed[ANSWER].

Promblem 1: A rocket has a built-in redundant system. In this system, if component $K_1$ fails, it is bypassed and component $K_2$ is used. If component $K_2$ fails, it is bypassed and component $K_3$ is used. (An example of a system with these kinds of components is three computer systems.) Suppose that the probability of failure of any one component is 0.15 , and assume that the failures of these components are mutually independent events. Let $A_i$ denote the event that component $K_i$ fails for $i=1,2,3$. What is the probability that the system fails?
Explanation for Problem 1: 
Because the system fails if $K_1$ fails and $K_2$ fails and $K_3$ fails, the probability that the system does not fail is given by
$$
\begin{aligned}
P\left[\left(A_1 \cap A_2 \cap A_3\right)^{\prime}\right] & =1-P\left(A_1 \cap A_2 \cap A_3\right) \\
& =1-P\left(A_1\right) P\left(A_2\right) P\left(A_3\right) \\
& =1-(0.15)^3 \\
& =0.9966 .
\end{aligned}
$$
Therefore, the answer is \boxed{0.9966}.

Promblem 2: At a county fair carnival game there are 25 balloons on a board, of which 10 balloons 1.3-5 are yellow, 8 are red, and 7 are green. A player throws darts at the balloons to win a prize and randomly hits one of them. Given that the first balloon hit is yellow, what is the probability that the next balloon hit is also yellow?
Explanation for Problem 2: Of the 24 remaining balloons, 9 are yellow, so a natural value to assign to this conditional probability is $9 / 24$.
Therefore, the answer is \boxed{0.375}.

Promblem 3: A certain food service gives the following choices for dinner: $E_1$, soup or tomato 1.2-2 juice; $E_2$, steak or shrimp; $E_3$, French fried potatoes, mashed potatoes, or a baked potato; $E_4$, corn or peas; $E_5$, jello, tossed salad, cottage cheese, or coleslaw; $E_6$, cake, cookies, pudding, brownie, vanilla ice cream, chocolate ice cream, or orange sherbet; $E_7$, coffee, tea, milk, or punch. How many different dinner selections are possible if one of the listed choices is made for each of $E_1, E_2, \ldots$, and $E_7$ ?
Explanation for Problem 3:  By the multiplication principle, there are
$(2)(2)(3)(2)(4)(7)(4)=2688$
different combinations.

Therefore, the answer is \boxed{2688}.

Promblem 4: A grade school boy has five blue and four white marbles in his left pocket and four blue and five white marbles in his right pocket. If he transfers one marble at random from his left to his right pocket, what is the probability of his then drawing a blue marble from his right pocket?
Explanation for Problem 4: For notation, let $B L, B R$, and $W L$ denote drawing blue from left pocket, blue from right pocket, and white from left pocket, respectively. Then
$$
\begin{aligned}
P(B R) & =P(B L \cap B R)+P(W L \cap B R) \\
& =P(B L) P(B R \mid B L)+P(W L) P(B R \mid W L) \\
& =\frac{5}{9} \cdot \frac{5}{10}+\frac{4}{9} \cdot \frac{4}{10}=\frac{41}{90}
\end{aligned}
$$
is the desired probability.
Therefore, the answer is \boxed{0.444444444444444 }.

Promblem 5: In an orchid show, seven orchids are to be placed along one side of the greenhouse. There are four lavender orchids and three white orchids. How many ways are there to lineup these orchids?
Explanation for Problem 5: Considering only the color of the orchids, we see that the number of lineups of the orchids is
$$
\left(\begin{array}{l}
7 \\
4
\end{array}\right)=\frac{7 !}{4 ! 3 !}=35 \text {. }
$$
Therefore, the answer is \boxed{35}.

