Isabel Briggs Myers was a pioneer in the study of personality types. The personality types are broadly defined according to four main preferences. Do married couples choose similar or different personality types in their mates? The following data give an indication.

Similarities and Differences in a Random Sample of 375 Married Couples

Number of Similar Preferences Number of Married CouplesAll four29Three126Two122One63None35

Suppose that a married couple is selected at random.

(a) Use the data to estimate the probability that they will have 0, 1, 2, 3, or 4 personality preferences in common. (For each answer, enter a number. Enter your answers to 2 decimal places.)01234

(b) Do the probabilities add up to 1? Why should they?

Yes, because they do not cover the entire sample space.

No, because they do not cover the entire sample space.

Yes, because they cover the entire sample space.

No, because they cover the entire sample space.

What is the sample space in this problem?

0, 1, 2, 3 personality preferences in common

1, 2, 3, 4 personality preferences in common

0, 1, 2, 3, 4, 5 personality preferences in common

0, 1, 2, 3, 4 personality preferences in common

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The following question involves a standard deck of 52 playing cards. In such a deck of cards there are four suits of 13 cards each. The four suits are: hearts, diamonds, clubs, and spades. The 26 cards included in hearts and diamonds are red. The 26 cards included in clubs and spades are black. The 13 cards in each suit are: 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, and Ace. This means there are four Aces, four Kings, four Queens, four 10s, etc., down to four 2s in each deck.

You draw two cards from a standard deck of 52 cards without replacing the first one before drawing the second.

(a) Are the outcomes on the two cards independent? Why?

No. The events cannot occur together.

No. The probability of drawing a specific second card depends on the identity of the first card.

Yes. The probability of drawing a specific second card is the same regardless of the identity of the first drawn card.

Yes. The events can occur together.

(b) Find P(ace on 1st card and queen on 2nd). (Enter your answer as a fraction.)

(c) Find P(queen on 1st card and ace on 2nd). (Enter your answer as a fraction.)

(d) Find the probability of drawing an ace and a queen in either order. (Enter your answer as a fraction.)

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What is the probability of the following. (For each answer, enter an exact number.)

(a) An event A that is certain to occur?

(b) An event B that is impossible?

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What is the income distribution of super shoppers? A supermarket super shopper is defined as a shopper for whom at least 70% of the items purchased were on sale or purchased with a coupon. In the following table, income units are in thousands of dollars, and each interval goes up to but does not include the given high value. The midpoints are given to the nearest thousand dollars.

Income range5-1515-2525-3535-4545-5555 or moreMidpoint x102030405060Percent of super shoppers20%15%21%17%20%7%

(a) Using the income midpoints x and the percent of super shoppers, do we have a valid probability distribution? Explain.

No. The events are indistinct and the probabilities sum to more than 1.

Yes. The events are indistinct and the probabilities sum to less than 1.

No. The events are indistinct and the probabilities sum to 1.

Yes. The events are distinct and the probabilities sum to 1.

Yes. The events are distinct and the probabilities do not sum to 1.

(b) Use a histogram to graph the probability distribution of part (a). (Select the correct graph.)

(c) Compute the expected income μ of a super shopper (in thousands of dollars). (Enter a number. Round your answer to two decimal places.)

μ = thousands of dollars

(d) Compute the standard deviation σ for the income of super shoppers (in thousands of dollars). (Enter a number. Round your answer to two decimal places.)

σ = thousands of dollars

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Compute C5,2. (Enter an exact number.)

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A particular lake is known to be one of the best places to catch a certain type of fish. In this table, x = number of fish caught in a 6-hour period. The percentage data are the percentages of fishermen who caught x fish in a 6-hour period while fishing from shore.

x01234 or more%43%35%15%6%1%

(a) Convert the percentages to probabilities and make a histogram of the probability distribution. (Select the correct graph.)

(b) Find the probability that a fisherman selected at random fishing from shore catches one or more fish in a 6-hour period. (Enter a number. Round your answer to two decimal places.)

(c) Find the probability that a fisherman selected at random fishing from shore catches two or more fish in a 6-hour period. (Enter a number. Round your answer to two decimal places.)

(d) Compute μ, the expected value of the number of fish caught per fisherman in a 6-hour period (round 4 or more to 4). (Enter a number. Round your answer to two decimal places.)

μ = fish

(e) Compute σ, the standard deviation of the number of fish caught per fisherman in a 6-hour period (round 4 or more to 4). (Enter a number. Round your answer to three decimal places.)

σ = fish

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Which of the following are continuous variables, and which are discrete?

(a) number of traffic fatalities per year in the state of Florida

continuous

discrete

(b) distance a golf ball travels after being hit with a driver

continuous

discrete

(c) time required to drive from home to college on any given day

continuous

discrete

(d) number of ships in Pearl Harbor on any given day

continuous

discrete

(e) your weight before breakfast each morning

continuous

discrete

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Compute P7,2. (Enter an exact number.)

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There are seven wires which need to be attached to a circuit board. A robotic device will attach the wires. The wires can be attached in any order, and the production manager wishes to determine which order would be fastest for the robot to use. Use the multiplication rule of counting to determine the number of possible sequences of assembly that must be tested. (Hint: There are seven choices for the first wire, six for the second wire, five for the third wire, etc. Enter an exact number.)

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The quality-control inspector of a production plant will reject a batch of syringes if two or more defective syringes are found in a random sample of eight syringes taken from the batch. Suppose the batch contains 5% defective syringes.

(a) Make a histogram showing the probabilities of r = 0, 1, 2, 3, …, 7 and 8 defective syringes in a random sample of eight syringes. (Select the correct graph.)

(b) Find μ (in terms of the number of syringes). (Enter a number. Enter your answer to two decimal places.)

μ = syringes

What is the expected number of defective syringes the inspector will find? (Enter a number. Enter your answer to two decimal places.)

syringes

(c) What is the probability that the batch will be accepted? (Enter a number. Round your answer to three decimal places.)

(d) Find σ (in terms of the number of syringes). (Enter a number. Round your answer to three decimal places.)

σ = syringes

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What is the law of large numbers?

As the sample size decreases, the relative frequency of outcomes gets closer to the theoretical probability of the outcome.

As the sample size increases, the cumulative frequency of outcomes gets closer to the theoretical probability of the outcome.

As the sample size increases, the relative frequency of outcomes gets closer to the theoretical probability of the outcome.

As the sample size increases, the relative frequency of outcomes moves further from the theoretical probability of the outcome.

If you were using the relative frequency of an event to estimate the probability of the event, would it be better to use 100 trials or 500 trials? Explain.

It would be better to use 500 trials, because the law of large numbers would take effect.

It would be better to use 100 trials, because the law of large numbers would take effect.

It would be better to use 500 trials, because 100 trials is always too small.

It would be better to use 100 trials, because 500 trials is always too big.

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Given P(A) = 0.4 and P(B) = 0.6, do the following. (For each answer, enter a number.)

(a) If A and B are independent events, compute P(A and B).

(b) If P(A | B) = 0.5, compute P(A and B).

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A recent survey of 1030 U.S. adults selected at random showed that 647 consider the occupation of firefighter to have very great prestige. Estimate the probability (to the nearest hundredth) that a U.S. adult selected at random thinks the occupation of firefighter has very great prestige. (Enter a number.)

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Sociologists say that 90% of married women claim that their husband’s mother is the biggest bone of contention in their marriages (sex and money are lower-rated areas of contention). Suppose that six married women are having coffee together one morning. Find the following probabilities. (For each answer, enter a number. Round your answers to three decimal places.)

(a) All of them dislike their mother-in-law.

(b) None of them dislike their mother-in-law.

(c) At least four of them dislike their mother-in-law.

(d) No more than three of them dislike their mother-in-law.

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For each of the following situations, explain why the combinations rule or the permutations rule should be used.

(a) Determine the number of different groups of 5 items that can be selected from 12 distinct items.

Use the permutations rule, since the number of arrangements within each group is of interest.

Use the combinations rule, since only the items in the group is of concern.

Use the permutations rule, since only the items in the group is of concern.

Use the combinations rule, since the number of arrangements within each group is of interest.

(b) Determine the number of different arrangements of 5 items that can be selected from 12 distinct items.

Use the combinations rule, since the number of arrangements within each group is of interest.

Use the permutations rule, since the number of arrangements within each group is of interest.

Use the permutations rule, since only the items in the group is of concern.

Use the combinations rule, since only the items in the group is of concern.

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Consider the probability distribution shown below.

x012P(x)0.750.200.05

Compute the expected value of the distribution. (Enter a number.)

Compute the standard deviation of the distribution. (Enter a number. Round your answer to four decimal places.)