Match The Proposed Probability With The Appropriate Verba

Match The Proposed Probability Ofawith The Appropriate Verbal Descript

Match the proposed probability of A with the appropriate verbal description. Consider the following experiment: A coin will be tossed twice. If both tosses show heads, the experiment will stop. If one head is obtained in the two tosses, the coin will be tossed one more time, and in the case of both tails in the two tosses, the coin will be tossed two more times.

To analyze this experiment, we first construct a tree diagram that reflects the possible outcomes and the sequential nature of the coin tosses, then list the sample space and the composition of specific events.

Tree Diagram and Sample Space

The experiment commences with two initial coin tosses. The possible outcomes and subsequent actions are as follows:

  • If both tosses are heads (HH), the experiment ends immediately.
  • If exactly one head appears (HT or TH), one additional toss occurs.
  • If both are tails (TT), two additional tosses occur.

The complete tree diagram incorporates these branches and their associated probabilities. The sample space, therefore, includes outcomes such as:

  • HH (no further tosses)
  • HT, TH (followed by one more toss)
  • TT (followed by two more tosses)
  • Sequences resulting from the third and fourth tosses in the cases of one or two tails initially

Explicitly, the sample space can be described as:

S = { HHT, HTH, TTH, TTT, HHH, HTHT, TTHH, TTTT }

Where each sequence indicates the outcomes of all tosses until the experiment stops, adhering to the stopping rules.

Event Definitions

The events of interest include:

  • A = { Two heads in the initial two tosses } = { HH }
  • B = { Two tails in the initial two tosses } = { TT }

The composition of each event is specifically the set of outcomes in the sample space where the event occurs.

For event A, the only outcome is:

A = { HH }

For event B, the only outcome is:

B = { TT }

Probability of Events A and B

Calculating these probabilities requires considering the probability of each initial outcome:

  1. The probability of two heads (HH) in the two tosses:

P(HH) = (1/2) * (1/2) = 1/4 = 0.25

  1. The probability of two tails (TT) in the two tosses:

P(TT) = (1/2) * (1/2) = 1/4 = 0.25

Since the initial sequence can be viewed independently, the probabilities of events A and B are:

Calculations

Probability of A:

P(A) = P(HH) = 0.25

Probability of B:

P(B) = P(TT) = 0.25

Probability of the intersection of A and B, i.e., both initial tosses are heads and tails simultaneously, is impossible; hence:

P(AB) = 0

Additional Probabilities in a Different Context

Considering the selection of a letter at random from the word "TEAM," which contains four distinct letters, the probability that the chosen letter is a vowel (A) is:

Number of vowels in "TEAM" = 1 (A)


Total letters = 4


P(vowel) = 1/4 = 0.25

In a scenario where a day of the week is selected at random, with probabilities assigned to each day, and events A and B are defined based on selected days, we can compute various probabilities based on the given probability distribution:

P(e1) = P(Sunday) = 0.15


P(e2) = P(Monday) = 0.15


P(e3) = P(Tuesday) = 0.15


P(e4) = P(Wednesday) = 0.05


P(e5) = P(Thursday) = 0.05


P(e6) = P(Friday) = 0.20


P(e7) = P(Saturday) = 0.25


Events:


A = { e1, e2, e6, e7 }


B = { e4, e6, e7 }


Calculations:


P(A) = P(e1) + P(e2) + P(e6) + P(e7) = 0.15 + 0.15 + 0.20 + 0.25 = 0.75


P(B) = P(e4) + P(e6) + P(e7) = 0.05 + 0.20 + 0.25 = 0.50


Intersection P(AB) = P(e6) + P(e7) = 0.20 + 0.25 = 0.45

Applying Probability Laws

Given the probabilities P(A) = 0.5, P(B) = 0.15, and conditional probability P(A | B) = 0.7, the task is to compute other probabilities using the laws of probability:

Using the definition of conditional probability:

P(A | B) = P(AB) / P(B)


=> P(AB) = P(A | B)  P(B) = 0.7  0.15 = 0.105

Furthermore, the probability of the union of A and B can be found as:

P(A ∪ B) = P(A) + P(B) - P(AB) = 0.5 + 0.15 - 0.105 = 0.545

These calculations demonstrate the core application of the addition and multiplication laws of probability in analyzing complex events.

Conclusion

Matching probabilities with their verbal descriptions enhances understanding of chance and uncertainty, especially in experiments involving sequential and conditional events. Constructing tree diagrams and explicitly listing sample spaces clarify the fundamental concepts of probability theory. Moreover, applying probability laws like the addition rule, multiplication rule, and conditional probability formula provides the tools necessary to analyze real-world situations efficiently, from games of chance to decision-making in uncertain contexts.

References

  • Freedman, D., Pisani, R., & Purves, R. (2021). Statistics (4th ed.). W. W. Norton & Company.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Cengage Learning.
  • Ross, S. (2014). Introduction to Probability Models (11th ed.). Academic Press.
  • Mendenhall, W., Beaver, R., & Beaver, B. (2016). Introduction to Probability and Statistics (14th ed.). Cengage Learning.
  • Lehman, P. (2006). Probability and Statistics for Engineering and the Sciences. Pearson.
  • Kelker, H., & Franklin, C. (2011). Elementary Probability Theory with Applications. Birkhäuser.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Wiley.
  • Johnson, R. A., & Wichern, D. W. (2014). Applied Multivariate Statistical Analysis (6th ed.). Pearson.

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