. Three Dice of a Kind

Consider the following game: You roll six 6-sided dice d1,…,d6 and you win if some number appears 3 or more times. For example, if you roll:

(3,3,5,4,6,6)

then you lose. If you roll

(4,1,3,6,4,4)

then you win.

1. What is the probability that you win this game?

According to a Gallup poll, fifty-one percent of US adults oppose hydraulic fracturing (fracking) as a means of increasing the production of natural gas and oil in the USA. You randomly select eight US adults. Find the probability that the number that oppose fracking for this purpose is

1. Exactly two

2. Less than five

3. At least three

You are ordering a hamburger and can get up to 88 toppings, but each topping can only be used once. You tell the cashier to surprise you with the toppings you get. What is the probability that you get 33 toppings? Express your answer as a fraction or a decimal number rounded to four decimal places.

Final exam scores in a Math class with large number of students have mean 145 and standard deviation 4.1. Provided the scores of this Final exam follow a normal distribution, what's the probability that a student scores below 145 OR above 157.3? Find the answer without using the LSND program. (Write the answer in decimals)

The mean number of words per minute (WPM) read by sixth graders is 83 with a standard deviation of 12 WPM. If 89 sixth graders are randomly selected, what is the probability that the sample mean would differ from the population mean by greater than 1.4 WPM? Round your answer to four decimal places.

1. A probability distribution shows the relative likelihood of observing any particular value. TRUE or FALSE

2. Which of the following distributions describes the number of times an event occurs during a given period of time or space?

Uniform distribution

Exponential distribution

Poisson distribution

Normal distribution

None of the answer choices is correct.

In a political science class there are 15 political science majors and 9 non-political science majors. 4 students are randomly selected to present a topic. What is the probability that at least 2 of the 4 students selected are political science majors? Express your answer as a fraction or a decimal number rounded to four decimal places.

You flip a coin, if it is heads you will have a good day and if it is tails you will have a bad day. There are 30 days in total.

(a) What is the expectation and variance of the number of times you will have a good day throughout this 30 day stretch?

(b) What is the probability that every day will be bad for all of the 30 days?

An on-line production of printed-circuit boards (PCBs) involves ten successive sequences (i = 1, 2, 3, …, 10) of component-placement/assembly on the board. In a quality-control effort, the board is tested for its integrity in each of ten successive assembly-tasks. It is presumed that the reliability of final, completed PCB depends on possible, failure-proneness associated with each (successive) assembly-task involved. Table given below shows the number of PCBs (n_i) per successive batch tested. Also shown in the table, is the random number of boards failed in the assembly-line (depicting the set of random variables, RV: x_i) in each of the ten successive batches tested.

Determine the following:

1. Most appropriate average performance measure expressed in terms of AM or HM or GM values of the failures involved: Why?
2. Expected mean: E[.], median, mode, variance and standard deviation (s) values of the RV
3. Skewness of the RV expressed in terms of Pearson-1 skewness coefficient; and, specify the type of skewness in terms of relevant parameters that decide the type (left or right-handed) of asymmetry (based on the positions of average measures marked)
4. Plot (discrete) probability density function (pdf) as a function of RV and construct the associated cdf graph. Marking the mode value (x_Mo) on the cdf plot, indicate the range of RV from {x_i = (x_Mo − s)} to {x_i = (x_Mo + s)} with respect to the mode and estimate approximate value of the associated cumulative probability in that range of RV
5. Using the cdf-plot, determine the cumulative probability of RV falling in excess of: x_i ≥ (E[.] + s) and decide on approximate value of this tail-end cumulative probability of the RV

(Answer hints: AM ≈ 35.5; E[.] ≈ 36.9; and, P_sk-1 ≈ − 0.094)

This is a standard deviation contest. You must choose four numbers from the whole numbers 0 to 10, with repeats allowed.

Step 1:
Choose four numbers that have the smallest possible standard deviation. What is the numerical value of s in this case? Give your answer as a whole number (meaning no decimal places).

Step 2:
Is there more than one possibility for choosing four numbers that have the smallest possible standard deviation?