Exam 5: Discrete Probability Distributions

A manufacturer of headache medicine claims it is 70 percent effective within a few minutes. That is, out of every 100 users 70 get relief within a few minutes. A group of 12 patients are given the medicine. If the claim is true, what is the probability that 8 have relief within a few minutes?

Chances are 50-50 that a newborn baby will be a girl. For families with five children, what is the probability that all the children are girls?

i. If we measure the weight of an eggnog carton, the variable is referred to as being a discrete random variable. ii. If we toss two coins and count the number of heads, there could be 0, 1, or 2 heads. Since the exact number of heads resulting from this experiment is due to chance, the number of heads appearing is a random variable. iii. A random variable may be either discrete or continuous.

When surveyed for brand recognition, 98% of consumers recognize Coke. A new survey of 800 randomly selected consumers is to be conducted. For such a group of 800, determine the mean and standard deviation for the number who recognize the Coke brand name. Considering as unusual a result that differs from the mean by more than two standard deviations, it (___ (is/is not) ___) unusual to get 790 consumers who recognize the Coke brand name.

When a household is randomly selected, the probability distribution for the number x of cars owned is as described in the accompanying table. Find the mean and standard deviation of the probability distribution.

A marketing solutions poll of mutual funds and fund owners asked fund owners what action they took after the September 11^th2001 market drop. Sixteen percent of respondents said they bought more funds. If 600 fund owners were polled, calculate the mean and standard deviation of the number of respondents who bought more funds. Considering as unusual a result that differs from the mean by more than two standard deviations, it (___ (is/is not) ___) unusual that in one of these polls of 600 fund owners, 100 respondents bought more mutual funds.

How is a Poisson distribution skewed?

(i. If $\pi$ = 1/3 and n = 900, the mean of this binomial distribution is 300. (ii) If n = 900 and $\pi$ = 1/3, the variance of this binomial distribution is 200. (iii) If $\pi$ = 1/5 and n = 100, the standard deviation of this binomial distribution is 16.

Judging from recent experience, 5 percent of the computer keyboards produced by an automatic, high-speed machine are defective. If six keyboards are randomly selected, what is the probability that more than 3 of the keyboards are defective?

(i. To construct a binomial distribution we need to know the total number of trials and the probability of a success. (ii) If n = 900 and $\pi$ = 1/3, the variance of this binomial distribution is 200. (iii) If $\pi$ = 1/5 and n = 100, the standard deviation of this binomial distribution is 20.

A multiple-choice test consists of six questions, each with A-E answers. If you guess the answer to each question, how many questions can you expect to get correct?

David's gasoline station offers 4 cents off per litre if the customer pays in cash and does not use a credit card. Past evidence indicates that 40% of all customers pay in cash. During a one-hour period twenty-five customers buy gasoline at this station. What is the probability that no more than twenty pay in cash?

The production department has installed a new spray machine to paint automobile doors. As is common with most spray guns, unsightly blemishes often appear because of improper mixture or other problems. A worker counted the number of blemishes on each door. Most doors had no blemishes; a few had one; a very few had two, and so on. The average number was 0.5 per door. The distribution of blemishes followed the Poisson distribution. Out of 10,000 doors painted, about how many would have no blemishes?

What is the only variable in the Poisson probability formula?

An insurance agent has appointments with four prospective clients tomorrow. From past experience the agent knows that the probability of making a sale on any appointment is 1 out of 5. Using the rules of probability, what is the likelihood that the agent will sell a policy to at least 3 of the 4 prospective clients?

When surveyed for brand recognition, 98% of consumers recognize Coke. A new survey of 800 randomly selected consumers is to be conducted. For such a group of 800, determine the mean and standard deviation for the number who recognize the Coke brand name. Considering as unusual a result that differs from the mean by more than two standard deviations, it (___ (is/is not) ___) unusual to get 775 consumers who recognize the Coke brand name.

A multiple-choice test consists of ten questions, each with A-E answers. If you guess the answer to each question, how many questions can you expect to get correct? Also find the standard deviation of the number of questions you can expect to get correct.

i. The probability of a particular outcome, designated X, must always be between 0 and 1 inclusive. ii. A random variable represents the outcomes of an experiment. iii. The mean of a probability distribution is referred to as its expected value.

(i. The binomial distribution and the Poisson distribution have two possible experimental outcomes. (ii) The Poisson distribution or, the law of improbable events, has negatively skewed shape. (iii) In a Poisson distribution each trail is independent.

i. A random variable may be either discrete or continuous. ii. If Unique Buying Services has 100 employees, there might be 0, 1, 2, 3 up to 100 employees absent on Monday. In this case, the day of the week is the random variable. iii. A discrete variable may assume fractional or decimal values, but they must have distance between them.