Assignment 2 Outcomes And Outcome Spaces For The Experiments

Assignment 2 Outcomes And Outcome Spacesfor The Experiments Defined I

Assignment 2: Outcomes and Outcome Spaces For the experiments defined in Questions 1-4, write the set of all possible outcomes (the outcome space), also note whether the outcomes are equally likely, and whether they are mutually exclusive. 1. Toss a coin three times and record heads or tails on each toss. 2. Assuming that automobile accidents occur randomly, record the number of accidents in NYC on the day you complete this homework. 3. For a randomly selected adult male, record his height to the nearest inch. 4. Inject a drug into 5 randomly selected diseased mice. At the end of one week, record the number cured out of 5. 5. How many outcomes are in an experiment consisting of one toss of a coin? Two tosses? N tosses? (Assume H or T is recorded for each toss.) 6. For Question 3 above, write the list of outcomes in the following events: A = (the selectee is less than 6’ tall) B=(the selectee is between 5 ‘ and 6’5†tall, inclusive) 7. For Question 4 above, write the list of outcomes in each event: A=(all mice are cured) B=(more than half the mice are cured) C=(an even number of mice are cured) 8. For Question 1 above, write the outcomes in the following events: A=(all tosses show the same face) B=(at least one toss is heads) C=(at most one toss is heads) 9. Calculate the probability of each event in Problem 8.

Sample Paper For Above instruction

Introduction

Understanding outcomes and outcome spaces in probability experiments is fundamental to grasping how randomness influences various scenarios. This paper systematically explores the outcome sets, probability calculations, and mutual exclusivity of events related to multiple experiments, including coin tosses, accident frequency, height measurements, and medical treatment outcomes in mice. By analyzing these examples, it highlights the core principles of probability theory and their practical applications.

Outcome Spaces Identification

1. Tossing a coin three times:

The experiment involves repeated trials with two possible outcomes: heads (H) or tails (T). The sample space, S, comprises all possible sequences of three tosses, which amount to 2^3 = 8 outcomes:

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.

All outcomes are equally likely assuming a fair coin, with each having a probability of 1/8.

2. Number of automobile accidents in NYC:

This experiment records the number of accidents, which could range from 0 upwards, depending on historical data and incident rates. Assuming a reasonable upper limit (e.g., 0-10 accidents), the outcome space is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Outcomes are not equally likely, as accidents tend to follow a Poisson distribution, and occur independently.

3. Height of a randomly selected adult male:

The outcome space consists of all possible recorded heights to the nearest inch. Typically, adult male heights range from about 5'0" to 7'0", so the outcome space is {60'', 61'', 62'', ..., 84''}. The outcomes are not equally likely, as height distributions are normally distributed.

4. Number of mice cured out of 5 after drug injection:

This experiment's outcomes are the integers from 0 to 5, representing the count of cured mice. The total outcome space contains 6 elements: {0,1,2,3,4,5}. Outcomes are not equally likely; the probabilities depend on drug efficacy and biological variability.

Number of Outcomes in Multiple Tosses and Event Outcomes

- For one coin toss, outcomes are 2: {H, T}.

- For two tosses, outcomes are 4: {HH, HT, TH, TT}.

- For N tosses, outcomes are 2^N, reflecting exponential growth as N increases.

Event outcomes in height measurements and mice treatment are specified as follows:

Height experiments:

- A = {outcomes where height

- B = {outcomes where 5’ and 6’5’'), i.e., {60'', 61'', 62'', 63'', 64'', 65'', 66'', 67'', 68'', 69'', 70'', 71'', 72'', 73'', 74'', 75'', 76'', 77'', 78'', 79'', 80'', 81'', 82'', 83'', 84''}

Mouse Cure Outcomes:

- A = {all mice are cured}: {5}

- B = {more than half are cured}: {3,4,5}

- C = {even number cured}: {0,2,4}

Coin Toss Outcomes:

- A = {all tosses show the same face}: {HHH, TTT}

- B = {at least one head}: all outcomes except TTT

- C = {at most one head}: {HTT, THT, TTH, TTT}

Probability Calculations of Events

For the coin toss experiment:

- P(A) = 2/8 = 1/4

- P(B) = outcomes with at least one head: 7/8

- P(C) = outcomes with at most one head: 3/8

These probabilities are derived assuming each toss is independent and fair, making outcomes equally likely within the experiment's sample space.

Conclusion

Analyzing outcomes and events in probability experiments reveals the structure of randomness in everyday scenarios. Whether dealing with coin flips or biological responses, identifying the outcome space and calculating probabilities enable more informed decision-making and risk assessment. These foundational concepts are vital in fields ranging from finance to engineering, underscoring their broad applicability and importance.

References

  1. Ross, S. M. (2014). A First Course in Probability. Pearson.
  2. Grinstead, C. M., & Snell, J. L. (2012). Introduction to Probability. American Mathematical Society.
  3. Mitzenmacher, M., & Upfal, E. (2005). Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press.
  4. Griffiths, M., & Tenenbaum, J. (2020). Fundamentals of Probability with Examples and Applications. Springer.
  5. Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.

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