Two fair, distinct dice are rolled. What is the probability

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Two fair, distinct dice are rolled

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Two fair, distinct dice are rolled. What is the probability that the first dice comes up 1 given that the sum on the two dice is 6?

Use this information for problems 2, 3, and 4: The following data on the marital status of 990 U.S. adults was found in Current Population Reports:

Single

M_1 Married

M_2 Widowed

M_3 Divorced

M_4 Total

Male

S_1 129 298 13 40 480

Female

S_2 104 305 57 44 510

Total 233 603 70 84 990

Find P(M_2 )

Find P(S_1 )

Find P(M_4∩S_2 )

Interpret your results.

(a) Find P(M_3?S_1 )

(b) Find P(S_2 ?| M_4)

4. Are M_1 and S_1 independent events? Justify your answer.

5. Two fair, distinct dice (one red and one green) are rolled. Let A be the event the red die comes up even and B be the event the sum on the two dice is eight. Are A,B independent events?

6. For the 107th Congress, 18.7% of the members were senators and 50% of the senators were Democrats. Using the multiplication rule, determine the probability that a randomly selected member of the 107th is a senator who is a Democrat.

7. According to the Current Population Reports, 51% of U.S. adults are women. Opinion Dynamics Poll published in USA Today shows that 32% of U.S. women and 55% of U.S. men believe in aliens. What percentage of U.S. adults believe in aliens?

8. According to the American Lung Association 7% of the population has lung disease. Of the people having lung disease 90% are smokers. Of the people not having lung disease 20% are smokers. What are the chances that a smoker has lung disease?

R part

9. Using R find the inverse of the matrix

A=(?(1&3&-2@2&-6&11@3&-3&9))

10. Using R, write the function

f(x)= √(3x^2+7x-1)

and compute its values for x=1,…,6

Suppose the body length of a certain species is normally distributed with mean 39.8 in and standard deviation 2.05 inches. What is the probability that a randomly selected member of this species will have a body length of at least 40 inches?

Suppose that every three months, on average, an earthquake occurs in a certain region. Assuming this is a Poisson process, what is the probability that the next earthquake occurs after three but before seven months?

Suppose that, on average, γ-particles emitted from a substance follow a Poisson process. Suppose moreover that four particles are emitted every second. What is the probability that will take at least 2 seconds before the next two γ-particles are emitted?

The proportion of stocks that will increase in value is a beta random variable with α=4 and β=3. What is the probability that at least 70% of the stocks will increase in value?

Let f(x,y)=A(x^2+y^2 ) in 0≤x≤1 and 0≤y≤3.

Determine the value of the constant A that makes f(x,y) a joint probability density function.

Compute P(X≤1/2,Y≥2).

Solve problem 1 using R. Then, graph the associated pdf.

Solve problem 2 using R. Then, graph the associated pdf.

Solve problem 3 using R. Then, graph the associated pdf.

Solve problem 4 using R. Then, graph the associated pdf.

Graph, using R, the following beta distributions:

α=0.5,β=0.5

α=5,β=2

α=2,β=4

α=2,β=2