Suppose Emerson loses 30% of all staring contests.

(a) What is the probability that

Emerson loses two staring contests in a row?

(b) What is the probability that

Emerson loses three staring contests in a row?

(c) When events are independent, their complements are independent as well. Use this result to determine the probability that

Emerson loses three staring contests in a row, but does not lose four in a row.

The service time in minutes from admit to discharge for ten patients seeking care in a hospital emergency department are 21, 136, 185, 156, 3, 16, 48, 28, 100, and 12. Construct a normal probability plot, an exponential probability plot, and a Weibull probability plot for these data in R. Based on the plots, which distribution is the best one to describe the outpatient service time?

Practice.
1. There are N men and M women in a group. Two random persons are selected. Calculate the
probabilities:
a) a man and a woman are selected;
b) two men are selected;
c) two women are selected.
2. If necessary, correct the mistakes in formulas:
a) ?(∅) = 1;
b) ?(? ∪ ?) = ?(?) ∙ ?(?), if ? ∩ ? ≠ ∅.
c) If ? ⊂ ?, then ?(?) > ?(?),
d) ?(Ω) = 1.

Theory. Explain what probability is, formula of classical probability. List properties of probability.

The grade appeal process at a university requires that a jury be structured by selecting five individuals randomly from a pool of eight students and thirteen faculty.​

(a) What is the probability of selecting a jury of all​ students?

(b) What is the probability of selecting a jury of all​ faculty?

(c) What is the probability of selecting a jury of three students and two ​faculty?

​​(Round to five decimal places as​ needed.)

True/False Questions (with explanations)

1. For a normal form game, a strategy is a probability distribution over the set of outcomes. (T/F)

2. For a mixed strategy Nash equilibrium of a normal form game, Bob is mixing over two actions A and B. Bob can change his payoff by reducing the probability of playing A and increasing the probability of playing B? (T/F)

Consider rolling two four sided dice. One die is blue and the other is green.

(a) State the sample space.

(b) Compute the probability of getting two even numbers.

(c) Compute the probability of getting an even sum if the blue die rolls a 2.

What is the probability of drawing a four or a heart when drawing a single card from a standard deck of cards.

Anticipated consumer demand at a restaurant for free-range steaks next month can be modeled by a normal random variable with mean 1,500 pounds and standard deviation 110 pounds.

a. What is the probability that demand will exceed 1,300 pounds?

b. What is the probability that demand will be between 1,400 and 1,600 pounds?

c. The probability is 0.15 that demand will be more than how many pounds?

Suppose that on average 4 tornados occur in 8 years in a county in Oklahoma. If the occurrence of tornados follows a Poisson process: (a) What is the probability that less than 6 years pass before 2 tornados occur (b) What is the probability that there will be at least one tornado next year? (c) What is the probability that the time between successive tornados is greater than 3 years?

in a certain college, 25% of the students failed mathematics,15% of the students failed chemistry and 10% of the students failed both mathematics and chemistry. A student is selected at random,
I)if he failed the exam, what is the probability that he failed mathematics?
ii) if he failed mathematics, what is the probability that he failed chemistry?
iii) what is the probability that he failed mathematics or chemistry?

USING EXCEL:

Consider the probability that no fewer than 75 out of 111 houses will lose power once a year. Assume the probability that a given house will lose power once a year is 98%

Specify whether the normal curve can be used as an approximation to the binomial probability by verifying the necessary conditions.

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