A wheel of fortune consists of five equally sized sectors. One of the sectors is labeled "0", one is labeled "1" and one is labeled "2"; the other two sectors are labeled "9".
(1a) [2 marks] The wheel of fortune is spun four times. Calculate the probability that the numbers 2, 0, 1 and 9 are obtained in the specified order.
(1b) [3 marks] The wheel of fortune is spun twice. Determine the probability that the sum of the numbers obtained is at least 11.
(2) [3 marks] The random variable $X$ can only take the values 1, 4, 9 and 16. It is known that $P ( X = 9 ) = 0.2$ and $P ( X = 16 ) = 0.1$ as well as the expected value $E ( X ) = 5$. Using an approach for the expected value, determine the probabilities $P ( X = 1 )$ and $P ( X = 4 )$.
(3) [2 marks] Given is a Bernoulli chain with length $n$ and success probability $p$. Explain that for all $k \in \{ 0 ; 1 ; 2 ; \ldots ; n \}$ the relationship $B ( n ; p ; k ) = B ( n ; 1 - p ; n - k )$ holds.
A company organizes trips with an excursion ship that has space for 60 passengers.

Given is a Bernoulli chain with length $n$ and success probability $p$. Explain that for all $k \in \{ 0 ; 1 ; 2 ; \ldots ; n \}$ the relationship $B ( n ; p ; k ) = B ( n ; 1 - p ; n - k )$ holds.
A company organizes trips with an excursion ship that has space for 60 passengers.
(1) [3 marks] Consider a trip where the ship is fully booked. Among the passengers are adults, teenagers and children. Half of the passengers eat ice cream during the trip, of the adults only one in three, of the teenagers and children 75\%. Calculate how many adults participate in the trip.
To participate in a trip, one must make a reservation in advance without having to pay the fare yet. Based on experience, some of the people with reservations do not appear for the trip. For the 60 available seats, the company therefore allows up to 64 reservations. It should be assumed that 64 reservations are actually made for each trip. If more than 60 people with reservations appear for the trip, only 60 of them can participate; the rest must be turned away. The random variable $X$ describes the number of people with reservations who do not appear for the trip. For simplicity, it should be assumed that $X$ is binomially distributed, where the probability that a randomly selected person with a reservation does not appear for the trip is 10\%. The table shown supplements the approved reference material.
Binomial distribution cumulative; $k \mapsto \sum _ { i = 0 } ^ { k } B ( n ; p ; i )$

n	k	$\mathrm { p } = 0.10$	$\mathrm { p } = 0.11$	$\mathrm { p } = 0.12$	$\mathrm { p } = 0.13$	$\mathrm { p } = 0.14$	$\mathrm { p } = 0.15$	$\mathrm { p } = 0.16$	p = 0.17
\multirow[t]{7}{*}{64}	0	0.00118	0.00058	0.00028	0.00013	0.00006	0.00003	0.00001	0.00001
	1	0.00956	0.00514	0.00272	0.00142	0.00073	0.00037	0.00019	0.00009
	2	0.03891	0.02290	0.01321	0.00748	0.00417	0.00228	0.00123	0.00065
	3	0.10629	0.06827	0.04277	0.02620	0.01572	0.00924	0.00533	0.00302
	4	0.22047	0.15377	0.10425	0.06886	0.04439	0.02797	0.01725	0.01043
	5	0.37271	0.28059	0.20485	0.14534	0.10040	0.06763	0.04450	0.02863
	...	...	...	...	...	...	...	...	...

(2a) [1 marks] Give a reason why the assumption that the random variable $X$ is binomially distributed is a simplification in the context of the problem.
(2b) [3 marks] Determine the probability that no person with a reservation needs to be turned away.
(2c) [3 marks] For the company, it would be helpful if the probability of having to turn away at least one person with a reservation were at most one percent. For this, the probability that a randomly selected person with a reservation does not appear for the trip would need to be at least a certain value. Determine this value to the nearest whole percent.
The company sets up an online portal for reservations and suspects that this could increase the proportion of people with reservations who do not appear for the respective trip. As a basis for deciding whether more than 64 reservations should be allowed per trip in the future, the null hypothesis "The probability that a randomly selected person with a reservation does not appear for the trip is at most 10\%." is to be tested using a sample of 200 people with reservations at a significance level of 5\%. Before the test is conducted, it is determined that the number of possible reservations per trip will only be increased if the null hypothesis would need to be rejected based on the test result.
(2d) [5 marks] Determine the associated decision rule.
(2e) [3 marks] Decide whether the choice of the null hypothesis was primarily motivated by the interest in having fewer empty seats or the interest in not having to turn away more people with reservations. Justify your decision.
(2f) [2 marks] Describe the associated Type II error and the resulting consequence in the context of the problem.