For full marks please remember to show all of your work and give a therefore statement.

Space, the final frontier! Your mission is to help the Starship Enterprise and its crew complete the following tasks.

1. During a visit to Sigma Iota II, Captain Kirk plays a game of cards known as Fizzbin.

Just like poker, 5 cards are dealt. The goal is to get a hand known as a Shronk. A Shronk has 2 cards from one suit and 3 cards from another suit. If you get a Shronk you win, otherwise, you lose 10 Federation Credits. What should you win in order to make this a fair game?

2. During a routine patrol, Enterprise receives a distress call from Earth outpost 3. It turns out that the Romulans have sent a Warbird to attack the outpost. In the resulting battle, Enterprise and the Romulan come to a point where the commanders can choose from two courses, Up or Down.

Capt. Kirk and the Romulan each play their strategies with a probability of 0.5.

a.) Make a distribution for the payoffs to Captain Kirk.

3. At a meeting of the Federation Council, there has been an outbreak of Rigellian Fever! Rigellian Fever can have up to 3 symptoms, headache, tiredness, and fever. Dr. McCoy has collected information on the number of people with these symptoms.

Symptom(s)	Frequency
headache	24
tiredness	27
fever	27
headache and fever	12
tiredness and fever	14
tiredness and headache	10
All 3	8

a.) How many delegates are at the Council?

b.) What is the probability that a delegate has a fever given that they have a headache?

c.)What is the probability that a delegate has either tiredness or headache?

4. Tribbles are very prolific creatures!

The number of young that a tribble gives birth to in a week is normally distributed with a mean of 58 young and a standard deviation of 4. Use the Empirical Rule to help answer this question.

a.) What is the probability that a tribble gives birth to more than 62 young? b.) What is the probability that a tribble gives birth to between 50 and 58 young?

5. Mr. Scott has collected some information about the 5 power couplings on the Enterprise. Due to a recent encounter with an Ion Storm, the couplings will reset their polarity every 10 minutes. The polarity can be either positive or negative and each polarity has the same probability of occurring in each power coupling (0.5 each). So long as 4 of 5 power couplings have the same, matching, polarity, Enterprise will have main power. Otherwise, Enterprise will have to use Auxiliary power. What is the probability that Enterprise will have main power the next time the power couplings reset?

6. During a recent series of 16 test detonations of photon torpedoes, the average yield of a photon torpedo was120 Isotons and the standard deviation is known to be 1.5 Isotons. Construct a 95% confidence interval estimator for the true mean yield of a photon torpedo. Provide an interpretation of the interval.

7. The Doomsday Machine has opened fire on Enterprise!

The probability that anyone blast from its weapon crashes through the Enterprises' shield is 0.2. If the Doomsday machine takes 5 shots at Enterprise, what is the probability that 2 of 5 shots crash through the shields?