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Upper level probability

### Let A represent having soup and let B represent having salad for lunch. Which statement is true? Having soup and salad for lunch are independent events because $$\displaystyle{P}{\left({A}∣{B}\right)}={P}{\left({A}\right)}{\quad\text{and}\quad}{P}{\left({B}∣{A}\right)}={P}{\left({B}\right)}$$. Having soup and salad for lunch are not independent because $$\displaystyle{P}{\left({A}∣{B}\right)}={P}{\left({A}\right)}{\quad\text{and}\quad}{P}{\left({B}∣{A}\right)}={P}{\left({B}\right)}$$. Having soup and salad are not independent events because $$\displaystyle{P}{\left({A}∣{B}\right)}≠{P}{\left({A}\right)}{\quad\text{and}\quad}{P}{\left({B}∣{A}\right)}≠{P}{\left({B}\right)}$$. Having soup and salad for lunch are independent events because $$\displaystyle{P}{\left({A}∣{B}\right)}≠{P}{\left({A}\right)}{\quad\text{and}\quad}{P}{\left({B}∣{A}\right)}≠{P}{\left({B}\right)}$$

Upper level probability

### When a British travelling company reduces fares of a particular trip from London to Nottingham. A small taxi can carry four passengers. The time between calls for tickets is exponentially distributed with a mean of 30 minutes. Assume that each call orders one ticket. What is the probability that the taxi is filled in less than 3 hours from the time of the fare reduction?

Upper level probability

### An analysis of variance of the yields of five different varieties of wheat, observed on one plot each at each of six different locations. The data from this randomized block design are listed here: $$\begin{array}{|c|c|} \hline Varieties & Location 1& Location 2& Location 3& Location 4& Location 5& Location 6 \\ \hline A& 35.3& 31.0& 32.7& 36.8& 37.2& 33.1 \\ \hline B& 30.7& 32.2& 31.4& 31.7& 35.0& 32.7\\ \hline C& 38.2& 33.4& 33.6& 37.1& 37.3& 38.2\\ \hline D& 34.9& 36.1& 35.2& 38.3& 40.2& 36.0\\ \hline E& 32.4& 28.9& 29.2& 30.7& 33.9& 32.1\\ \hline \end{array}$$ a. Use the appropriate nonparametric test to determine whether the data provide sufficient evidence to indicate a difference in the yields for the five different varieties of wheat. Test using $$\alpha\alpha=.05$$. b. How do the analysis of variance F test compare with the test in part a? Explain.

Upper level probability

### From a standard 52-card deck, what is the probability that a drawn card is a club or anything besides a face card?

Upper level probability

### Assuming the null hypothesis is true, what is the probability that our z-test statistic would fall outside the z-critical boundaries (in the tails of the distribution the region of rejection) for anв $$\alpha=0.05?$$

Upper level probability

### $$\frac{8!}{(5!\times3!)}$$

Upper level probability

### Random variables are represented by ___-case letters, while particular values of random variables are represented by ___-case letters.

Upper level probability

### The area between $$z=0$$ and $$z= 1.1$$ under the standard nomad curve is ?

Upper level probability

### Find the probability density function of $$Y=e^{X}$$, when X is normally distributed with parameters $$\mu\ \text{and}\ \sigma^{2}$$. The random variable Y is said to have a lognormal distribution (since log Y has a normal distribution) with parameters $$\mu\ \text{and}\ \sigma^{2}$$

Upper level probability

### A card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of drawing a face card or a 5?

Upper level probability

### Select the best answer. Which of the following random variables is geometric? (a) The number of times I have to roll a die to get two 6s. (b) The number of cards I deal from a well-shuffled deck of 52 cards until I get a heart. (c) The number of digits I read in a randomly selected row of the random digits table until I find a 7. (d) The number of 7s in a row of 40 random digits. (e) The number of 6s I get if I roll a die 10 times.

Upper level probability

### A population of values has a normal distribution with $$\displaystyle\mu={198.8}$$ and $$\displaystyle\sigma={69.2}$$. You intend to draw a random sample of size $$\displaystyle{n}={147}$$. Find the probability that a sample of size $$\displaystyle{n}={147}$$ is randomly selected with a mean between 184 and 205.1. $$\displaystyle{P}{\left({184}{<}{M}{<}{205.1}\right)}=$$? Write your answers as numbers accurate to 4 decimal places.

Upper level probability

### A shipment of 12 microwave ovens contains three defective units. A vending company has ordered four of these units, and because all are packaged identically, the selection will be random. What is the probability that: a) All four units are good? b) Exactly two units are good? c) At least two units are good?

Upper level probability

### There are 11 players in team A. It is known that four of them (call them high scoring) score a penalty with probability 0.8 each and the other seven (low-scoring) with probability 0.5 each. There is a penalty shoot-out with another team, team B (of 11 players), in which every player from each team takes exactly one shot, so that 22 shots are to be taken. Every player in team B scores with probability 0.64. The team that scores the most goals wins. Which team is expected to win? Justify your answer

Upper level probability

### What is the probability of tossing at least 1 tail if you toss 3 coins at once?

Upper level probability

### For any three events A,B, and D, such that P(D) >0, prove that $$\displaystyle{P}{\left({A}\cup{B}{\mid}{D}\right)}={P}{\left({A}{\mid}{D}\right)}+{P}{\left({B}{\mid}{D}\right)}-{P}{\left({A}\cap{B}{\mid}{D}\right)}$$.

Upper level probability

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