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Exam 5: Probability Distributions

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Question 41

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Find the mean, µ, for the binomial distribution which has the stated values of n and p. Round answer to the nearest tenth - $n = 20 ; p = 3 / 5$

A) $\mu = 12.3$
B) $\mu = 12.0$
C) $\boldsymbol { \mu } = 11.5$
D) $\boldsymbol { \mu } = 12.7$

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Question 42

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Solve the problem. -The probability of winning a certain lottery is 1/56,728. For people who play 517 times, find the standard deviation for the number of wins.

A) 0.0955
B) 0.1046
C) 0.0091
D) 2.1706

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Question 43

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The binomial distribution applies only to cases involving two types of outcomes, whereas the multinomial distribution involves more than two categories. Suppose we have three types of mutually exclusive outcomes denoted by A, B, and C. Let P A = p1, P B = p2, P C = p3. In n independent trials, the probability of x1 outcomes of type $A , x _ { 2 }$ outcomes of type B, and x3 outcomes of type C is given by $\frac { \mathrm { n } ! } { \left( \mathrm { x } _ { 1 } \right) ! \left( \mathrm { x } _ { 2 } \right) ! \left( \mathrm { x } _ { 3 } \right) ! } \cdot \mathrm { p } _ { 1 } \mathrm { x } _ { 1 } \cdot \mathrm { p } _ { 2 } ^ { \mathrm { x } _ { 2 } } \cdot \mathrm { p } _ { 3 } ^ { \mathrm { x } _ { 3 } }$ A genetics experiment involves four mutually exclusive genotypes identified as A, B, C, and D, and they are all equally likely. If 13 offspring are tested, find the probability of getting exactly 2 A's,4 B's,2 C's, and 5 D's by expanding the above expression so that it applies to four types of outcomes instead of only three.

Solve the problem. -The probability that a radish seed will germinate is 0.7. A gardener plants seeds in batches of 13. Find the standard deviation for the number of seeds germinating in each batch.

Find the mean of the given probability distribution. -The accompanying table shows the probability distribution for x, the number that shows up when a loaded die is rolled. $\begin{array} { c | c } \mathrm { x } & \mathrm { P } ( \mathrm { x } ) \\\hline 1 & 0.13 \\2 & 0.12 \\3 & 0.16 \\4 & 0.13 \\5 & 0.15 \\6 & 0.31\end{array}$

Solve the problem. -The probabilities that a batch of 4 computers will contain 0, 1, 2, 3, and 4 defective computers are 0.6274, 0.3102, 0.0575, 0.0047, and 0.0001, respectively. Find the standard deviation for the probability distribution.

Focus groups of 12 people are randomly selected to discuss products of the Yummy Company. It is determined that the mean number (per group) who recognize the Yummy brand name is 9.4, and the standard deviation is 0.98. Would it be unusual to randomly select 12 people and find that fewer than 5 recognize the Yummy brand name?

Answer the question. -Choosing 5 people (without replacement) from a group of 59 people, of which 15 are women, keeping track of the number of men chosen.

Assume that there is a 0.05 probability that a sports playoff series will last four games, a 0.45 probability that it will last five games, a 0.45 probability that it will last six games, and a 0.05 probability that it will last seven games. Is it unusual for a team to win a series in 7 games?

Find the mean, µ, for the binomial distribution which has the stated values of n and p. Round answer to the nearest tenth - $\mathrm { n } = 665 ; \mathrm { p } = .7$

The number of lightning strikes in a year at the top of a particular mountain has a Poisson distribution with a mean of 3.9. Find the probability that in a randomly selected year, the number of lightning strikes is 2.

Provide an appropriate response. -Do probability distributions measure what did happen or what will probably happen? How do we use probability distributions to make decisions?

Solve the problem. -According to a college survey, 22% of all students work full time. Find the mean for the number of students who work full time in samples of size 16.

Use the given values of n and p to find the minimum usual value µ - 2? and the maximum usual value µ + 2? -n = 94, p = 0.20

Use the normal distribution to approximate the desired probability. -Find the probability of selecting 9 or more girls.

Provide an appropriate response. -Describe the differences in the Poisson and the binomial distribution.

The Columbia Power Company experiences power failures with a mean of µ = 0.210 per day. Find the probability that there are exactly two power failures in a particular day.

Provide an appropriate response. -List the four requirements for a binomial distribution. Describe an experiment which is binomial and discuss how the experiment fits each of the four requirements.

Assume that x is a random variable in a probability distribution with mean µ and standard deviation ?. Find expressions for the mean and standard deviation if every value of x is modified by first being multiplied by 3, then increased by 5.

A machine has 9 identical components which function independently. The probability that a component will fail is 0.2. The machine will stop working if more than three components fail. Find the probability that the machine will be working.