Assistance for AMU MATH 120 Intro to statistics Homework week 3 American Military University is available at Domyclass

1. 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?

2.

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.

3. A recent survey of 1030 U.S. adults selected at random showed that 625 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.)

4. Consider a family with 4 children. Assume the probability that one child is a boy is 0.5 and the probability that one child is a girl is also 0.5, and that the events “boy” and “girl” are independent.

(a) List the equally likely events for the gender of the 4 children, from oldest to youngest. (Let M represent a boy (male) and F represent a girl (female). Select all that apply.)

- FMMM
- FFFF
- MFFM
- FFMF
- MFMF
- FFFM
- three M’s, one F
- MMMM
- MMMF
- one M, three F’s
- FMFM
- FMFF
- MMFF
- two M’s, two F’s
- MMFM
- FMMF
- MFMM
- FFMM
- MFFF

(b) What is the probability that all 4 children are male? (Enter your answer as a fraction.)

Notice that the complement of the event “all four children are male” is “at least one of the children is female.” Use this information to compute the probability that at least one child is female. (Enter your answer as a fraction.)

5. Consider the following.

(a) Explain why −0.41 cannot be the probability of some event.

- A probability must be between zero and one.
- A probability must be greater than one.
- A probability must be an integer.

(b) Explain why 1.21 cannot be the probability of some event.

- A probability must be between zero and one.
- A probability must be an integer.
- A probability must be negative.

(c) Explain why 120% cannot be the probability of some event.

- A probability must be between zero and one.
- A probability must be an integer.
- A probability must be negative.

(d) Can the number 0.56 be the probability of an event? Explain.

- No, this is too large to be a probability.
- No, a probability cannot be positive.
- Yes, it is a number between 0 and 1.
- Yes, it is a number between 0.5 and 1.

6. 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 Couples All four 31 Three 135 Two 120 One 67 None 22 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.)

0 1 2 3 4

(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

7. If two events are mutually exclusive, can they occur concurrently? Explain.

- Yes. By definition, mutually exclusive events can occur together.
- No. By definition, mutually exclusive events cannot occur together.
- No. Two events will never occur concurrently.
- Yes. Any two events can occur concurrently.

8. Given P(A) = 0.5 and P(B) = 0.2, do the following. (For each answer, enter a number.)

(a) If A and B are mutually exclusive events, compute P(A or B).

(b) If P(A and B) = 0.1, compute P(A or B).

9. Given P(A) = 0.6 and P(B) = 0.2, 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).

10. 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?

- 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.

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

- No. The events cannot occur together.

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

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

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

11. You draw two cards from a standard deck of 52 cards, but before you draw the second card, you put the first one back and reshuffle the deck.

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

- 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.

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

- No. The events cannot occur together.

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

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

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

12. Diagnostic tests of medical conditions can have several types of results. The test result can be positive or negative, whether or not a patient has the condition. A positive test (+) indicates that the patient has the condition. A negative test (−) indicates that the patient does not have the condition. Remember, a positive test does not prove the patient has the condition. Additional medical work may be required. Consider a random sample of 200 patients, some of whom have a medical condition and some of whom do not. Results of a new diagnostic test for a condition are shown.

Condition Present Condition Absent Row Total

Test Result + 102 28 130 Test Result − 21 49 70 Column Total 123 77 200 Assume the sample is representative of the entire population. For a person selected at random, compute the following probabilities. (Enter your answers as fractions.)

(a) P(+ | condition present); this is known as the sensitivity of a test.

(b) P(− | condition present); this is known as the falsenegative rate.

(c) P(− | condition absent); this is known as the specificity of a test.

(d) P(+ | condition absent); this is known as the falsepositive rate.

(e) P(condition present and +); this is the predictive value of the test.

(f) P(condition present and −).

13.

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 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.

(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 only the items in the group is of concern.

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

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

14. There are nine 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 nine choices for the first wire, eight for the second wire, seven for the third wire, etc. Enter an exact number.)

17 . One professor grades homework by randomly choosing 5 out of 12 homework problems to grade.

(a) How many different groups of 5 problems can be chosen from the 12 problems? (Enter an exact number.)

groups

(b) Probability extension: Jerry did only 5 problems of one assignment. What is the probability that the problems he did comprised the group that was selected to be graded? (Enter a number. Round your answer to four decimal places.)

(c) Silvia did 7 problems. How many different groups of 5 did she complete? (Enter an exact number.)

groups

What is the probability that one of the groups of 5 she completed comprised the group selected to be graded? (Enter a number. Round your answer to four decimal places.)