Question 1: An Experiment Consists Of Tossing A Coin Rolling ✓ Solved

Question 1an Experiment Consists Of Tossing A Coin Rolling A Die And

Describe an appropriate sample space for the experiment where a coin is tossed and a die is rolled.

Options include:

  • a. {(1, H, T), (2, H, T), (3, H, T), (4, H, T), (5, H, T), (6, H, T)}
  • b. {(H, 1), (T, 2), (H, 3), (T, 4), (H, 5), (T, 6), (T, 1), (T, 4), (T, 5), (H, 6)}
  • c. {(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)}
  • d. {(1, 2, 3, 4, 5, 6), (H, T)}
  • e. {(H, H, H, H, H, H, T, T, T, T, T, T, 1, 2, 3, 4, 5, 6)}

Choose the option that best describes the sample space for the experiment involving tossing a coin and rolling a die.

Sample Paper For Above instruction

The experiment involving tossing a coin and rolling a die requires identifying all possible outcomes that can occur. When a coin is tossed, there are two possible outcomes: heads (H) or tails (T). When a die is rolled, there are six possible outcomes: numbers 1 through 6. The total sample space is the Cartesian product of these two outcomes, pairing each coin result with each die outcome. Therefore, the sample space consists of 12 ordered pairs, each representing a specific combination of coin toss and die roll.

The appropriate sample space is the set of all ordered pairs where the first element indicates the coin result, and the second element indicates the die number. Explicitly, the sample space includes outcomes such as (H, 1), (H, 2), ..., (H, 6), and (T, 1), (T, 2), ..., (T, 6). This comprehensive set captures all possible outcomes of the experiment.

Option c accurately reflects this, listing all pairs with the coin result (H or T) paired with each die number (1 through 6). It accounts for all 12 possible outcomes and correctly models the sample space of the combined experiment.

Question 2

A sample of two transistors taken from a local electronics store was examined to determine whether the transistors were defective (d) or non-defective (n). What is an appropriate sample space for this experiment?

  • a. {dd, nn, nd}
  • b. {dd, dn, nd}
  • c. {dd, dn, nd, nn}
  • d. {(d, d), (n, n)}

Sample Paper For Above instruction

The experiment involves selecting two transistors and observing whether each is defective (d) or non-defective (n). The sample space comprises all ordered pairs where each element represents the condition of a transistor. Since there are two transistors, each can be either d or n, and the order matters because the first and second transistors are distinct.

All possible outcomes are: (d, d), (d, n), (n, d), and (n, n). These outcomes reflect every combination of defect status for the two transistors. It is important to include all these scenarios to correctly model the sample space, especially when analyzing probabilities related to defects.

Option d correctly lists the sample space as the set of ordered pairs with each transistor's defect status. It captures all four possible arrangements and is the most comprehensive and precise choice for this experiment.

Question 3

Eight players, A, B, C, D, E, F, G, and H, are competing in a series of elimination matches in a tennis tournament, where the winner of each preliminary match advances to the semifinals. The outline of the scheduled matches follows. Describe a sample space listing the possible participants in the finals.

  • a. S = {(A, D), (A, F), (A, H), (C, B), (C, F), (C, H), (E, B), (E, D), (E, H), (G, B), (G, D), (G, F)}
  • b. S = {(A, B), (A, D), (A, F), (A, H), (C, B), (C, D), (C, F), (C, H), (E, B), (E, D), (E, F), (E, H), (G, B), (G, D), (G, F), (G, H)}
  • c. S = {(A, E), (A, F), (A, G), (A, H), (B, E), (B, F), (B, G), (B, H), (C, E), (C, F), (C, G), (C, H), (D, E), (D, F), (D, G), (D, H)}
  • d. S = {(A, B), (A, C), (A, D), (B, C), (B, D), (C, D), (E, F), (E, G), (E, H), (F, G), (F, H), (G, H)}
  • e. None of the above.

Sample Paper For Above instruction

The experiment involves determining all possible pairs of finalists in a tennis tournament where eight players compete through elimination rounds. The goal is to list all unique pairs that could emerge as finalists. Each pair in the sample space signifies two players who could potentially compete in the final match.

To construct the sample space, it's necessary to consider that any two players could be the finalists, but since the matches are arranged in a specific bracket, only particular pairings are feasible based on the scheduled matches and possible outcome paths. The listed options represent different possible subsets of finalists, with options (a) and (b) listing various pairings based on potential matchups, but only one represents the complete set relevant to the question.

Option a lists 12 pairs, which might correspond to specific plausible matchups based on the tournament bracket. The sample space should include all unique pairs of finalists that can result from the tournament's structure without repetition and considering the possible pathways of advancement.

Therefore, the most comprehensive and relevant option for this context is option a, which accurately captures the set of all plausible finalists based on the tournament structure.

Question 4

An experiment consists of selecting a card from a standard deck of playing cards and noting whether the card is black (B) or red (R). What are the events of this experiment?

  • a. {B}, {R}, {B, B}, {R, R}
  • b. {B, R}, {R, B}
  • c. {∅, {B, R}, {R, B}, {R, R}, {B, B}}
  • d. {∅, {B}, {R}}
  • e. {∅, {B}, {R}, {B, R}}

Sample Paper For Above instruction

The experiment involves selecting one card randomly from a standard deck and observing its color, either black (B) or red (R). The events in the context of probability are the various outcomes or groups of outcomes that can occur during this experiment.

Each event represents a subset of the sample space of possible outcomes. For a single card drawn from the deck, there are two fundamental events: drawing a black card or a red card. These are mutually exclusive and collectively exhaustive events, meaning one of them must occur, but both cannot happen simultaneously.

Further, the sets like {B, B} or {R, R} do not add additional meaningful outcomes but represent duplicate events or combined events, which are typically not considered distinct in probability sets. The empty set ∅ signifies an impossible event, and the set {B, R} or {∅, {B, R}, {R, B}, ...} indicates combined outcomes or collections of events.

Thus, the standard and most appropriate way to represent the fundamental events of this experiment is {∅, {B}, {R}}, corresponding to no outcome and the basic events of drawing a black or a red card.

Therefore, option d is the most accurate, representing the sample space for the fundamental events of selecting a card's color.

Question 5

Let the sample space be outcomes p, q, r. List all events of this experiment.

  • a. ∅, {p}, {q}, {r}, {p, q}, {p, r}, {q, r}, S
  • b. {p}, {q}, {r}, {p, q}, {p, r}, {q, r}
  • c. ∅, {p}, {q}, {r}, {p, q}, {p, r}, {q, r}
  • d. {p}, {q}, {r}

Sample Paper For Above instruction

In probability theory, the events of an experiment are represented as subsets of the sample space, which includes all possible outcomes. For outcomes labeled p, q, and r, the events are all possible subsets of these outcomes, including the empty set, singletons, pairs, and the entire set.

Specifically, the events include:

  • the empty set ∅ (impossible event),
  • single outcomes {p}, {q}, {r},
  • pairs of outcomes {p, q}, {p, r}, {q, r},
  • the entire sample space S = {p, q, r}.

In set notation, the total events are the power set of the sample space, including all the above. The key is to include all possible combinations of outcomes, which are represented by all subsets of {p, q, r}.

Option a correctly lists all these subsets, making it the comprehensive answer for the events of this experiment.

Question 6

The customer service department of Universal Instruments conducted a survey among customers who had returned their purchase registration cards. Purchasers of its deluxe model home computer reported the length of time (t) in days before service was required.

Describe a sample space corresponding to this survey.

  • a. {t | 100 ≤ t ≤ 370}
  • b. {t | 0
  • c. {t | t > 0}
  • d. {t | 0

Sample Paper For Above instruction

The survey aims to collect data on the time (t) in days before the computer requires servicing. The sample space should encompass all possible durations in days that customers might report, considering the realistic range inferred from the context.

Option a specifies the set of all values t where t is between 100 and 370 days inclusive; this corresponds to typical expected service times, assuming minimal service is needed after 100 days and not exceeding 370 days.

Option b includes all t greater than zero up to 370 days, representing all plausible service durations without restriction, assuming service could be needed at any point within that period.

Option c extends the range to any positive value, including potentially very large times, which might be more general but less precise.

Option d restricts to times between 0 and 100 days, which seems too narrow given the context.

Therefore, the most appropriate sample space covers the range from just above zero to 370 days, accepting any value within that interval. Option b correctly captures this.

Question 7

Let the sample space be outcomes and let E and F be events of this experiment. Find the events E ∪ F and E ∩ F.

  • a. (Provide specific options here)
  • b. (Provide specific options here)
  • c. (Provide specific options here)
  • d. (Provide specific options here)

Sample Paper For Above instruction

In set theory, the union E ∪ F of two events includes all outcomes that are in either E, F, or both. The intersection E ∩ F includes only outcomes that are common to both E and F. To find these, one must know the specific outcomes belonging to E and F.

Without specific outcome sets, we generally define:

  • E ∪ F = {x | x ∈ E or x ∈ F}
  • E ∩ F = {x | x ∈ E and x ∈ F}

These set operations help in calculating probabilities of combined events, such as the probability that either event occurs (union) or both occur simultaneously (intersection). The precise outcomes depend on the given sets E and F.

Question 8

In a television game show, the winner is asked to select three prizes from five different prizes: A, B, C, D, and E. Describing a sample space of possible outcomes (order is not important), determine the number of points in the sample space corresponding to a selection that includes A.

  • a. 6
  • b. 2
  • c. 9
  • d. 4

Sample Paper For Above instruction

The problem involves selecting three prizes from five options without regard to order, with the additional condition that prize A is included in each selection. The total number of possible combinations that include A can be calculated by fixing A in each subset and choosing the remaining two prizes from the other four options (B, C, D, E).

The number of combinations choosing 2 from 4 is given by the binomial coefficient C(4, 2) = 6. Therefore, there are six different possible selections that include A: {A, B, C}, {A, B, D}, {A, B, E}, {A, C, D}, {A, C, E}, {A, D, E}.

This confirms that the number of possible outcomes with A included is 6, making option a the correct answer.

Question 9

Human blood is classified by the presence or absence of antigens A, B, and Rh. The classifications include AB+, AB-, O+, O-, and others. Using this information, determine the sample space corresponding to these blood groups.

  • a. {AB+, AB-, AO+, BO+, AO-, BO-, O+, O-}
  • b. {AB+, AB-, O+, O-}
  • c. {AB+, AB-, A+, B+, A-, B-, O+, O-, ABO-, AO+, AO-, BO+, BO-}
  • d. {AB+, AB-, A+, B+, A-, B-, O+, O-}
  • e. {A+, B+, A-, B-, O+, O-}

Sample Paper For Above instruction

Blood groups are classified based on the presence of antigens A, B, and Rh factor. The comprehensive sample space must include all possible combinations of these antigens and Rh status.

every blood group with different antigen and Rh combinations is represented in the sample space. For instance, blood type AB+ contains A, B, and Rh antigens; O- contains none of these antigens, and so forth. Enumeration includes all combinations such as AB+, AB-, A+, B+, A-, B-, O+, O-.

Option c encompasses all these possible classifications, capturing the complete set of blood groups based on antigen presence and Rh factor, thus providing an exhaustive sample space for the experiment.

Question 10

An experiment involves selecting a card from a standard deck and noting whether the card is black (B) or red (R). Describe an appropriate sample space for this experiment.

  • a. {R, R}
  • b. {R, B, B, R}
  • c. {B, B}
  • d. {BR, RB}
  • e. {B, R}

Sample Paper For Above instruction

The experiment involves choosing a single card from a deck and observing its color. The sample space includes all possible outcomes of this process, which are simply the colors the card might be.

The fundamental outcomes are 'Black' (B) and 'Red' (R). The sample space should reflect these two possibilities, representing all possible outcomes with no repetitions or combinations beyond these two outcomes.

Therefore, the appropriate sample space is the set {B, R}, which includes all fundamental events of drawing either a black or red card. The other options introduce unnecessary duplicates or combined representations not suitable for the basic outcomes.

Question 11

Let the sample space be outcomes, and let E and F be events of this experiment. Are the events E and F mutually exclusive?

  • a. no

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