A Dart Player Throws 10 Darts At The Dart Board

A Dart Player Throws 10 Darts At The Dart Board He Has A 20

1 A Dart Player Throws 10 Darts At The Dart Board He Has A 20

The assignment involves analyzing probability scenarios related to binomial and other probability distributions. Specifically, it covers calculating probabilities for a dart player’s success rate, understanding the appropriateness of different distributions in a team selection context, and modeling and computing probabilities concerning the count of sidewalk cracks observed by a child.

Paper For Above instruction

This paper explores key probability models and their applications in real-world scenarios, focusing on binomial distributions and alternative models suited for particular situations. The discussion begins with a probability problem involving a dart game, then examines the suitability of the binomial distribution in a team selection case, and concludes with modeling the occurrence of sidewalk cracks using an appropriate distribution.

Probability of Exact Successes in a Binomial Context

The first scenario involves a dart player throwing 10 darts with a 20% chance of hitting the bull’s eye each time. Assuming each throw is independent, the probability of exactly two successful hits can be modeled using the binomial distribution. The binomial probability mass function (PMF) is expressed as:

P(X=k) = C(n, k)  p^k  (1-p)^{n-k}

where n=10 (number of trials), k=2 (number of successes), and p=0.2 (probability of success per trial). Therefore, the probability that the player hits the bull's eye exactly twice is:

P(X=2) = C(10, 2)  (0.2)^2  (0.8)^8

Calculating the combination yields C(10, 2) = 45, and computing gives:

P(X=2) = 45  0.04  0.16777 ≈ 45 * 0.006711 ≈ 0.302

Thus, there is approximately a 30.2% chance that the dart player hits the bull’s eye exactly twice in 10 throws.

Team Selection and Distribution Considerations

The second scenario involves selecting 5 players out of 20, with 7 upperclassmen and 13 lowerclassmen. The question examines the appropriateness of using the binomial distribution and then computes the probability that exactly three of the starters are upperclassmen.

a. The binomial distribution assumes independent trials with a fixed probability of success (being an upperclassman). However, because the players are selected without replacement from a finite population, the trials are dependent; selecting one upperclassman reduces the number remaining in the pool and thus affects the probabilities for subsequent selections. Therefore, the binomial distribution is not strictly appropriate here.

b. Instead, the hypergeometric distribution should be used, as it models the probability of k successes (upperclassmen) in n draws without replacement from a finite population.

The hypergeometric probability mass function is:

P(X=k) = [C(K, k) * C(N - K, n - k)] / C(N, n)

Where:

- N=20 (total players),

- K=7 (number of upperclassmen),

- n=5 (number of starters),

- k=3 (number of upperclassmen among starters).

Calculating:

P(X=3) = [C(7, 3) * C(13, 2)] / C(20, 5)

C(7, 3) = 35, C(13, 2) = 78, and C(20, 5) = 15504.

Hence:

P(X=3) = (35 * 78) / 15504 ≈ 2730 / 15504 ≈ 0.176

Therefore, the probability that exactly three of the five starters are upperclassmen is approximately 17.6%. This calculation reflects the dependent nature of the selection process and confirms the need for the hypergeometric distribution.

Modeling Crack Counts in Sidewalks

The child counts sidewalk cracks over roughly 1/8 mile of sidewalk, with an expected value of 2 cracks. To choose the appropriate distribution:

a. Suitable Distribution

Given that cracks occur randomly and independently along the sidewalk segment, and the average number of cracks is small, the Poisson distribution is well-suited for modeling this scenario. The Poisson distribution describes the probability of a given number of events (cracks) in a fixed interval or space, assuming events occur independently at a constant average rate.

b. Probability of Observing 3 or More Cracks

Using Poisson distribution with λ=2:

P(X ≥ 3) = 1 - P(X ≤ 2) = 1 - [P(0) + P(1) + P(2)]


P(0) = e^{-2} * 2^{0} / 0! = e^{-2} ≈ 0.1353


P(1) = e^{-2} * 2^{1} / 1! ≈ 0.2707


P(2) = e^{-2} * 2^{2} / 2! ≈ 0.2707

Summing these:

P(0) + P(1) + P(2) ≈ 0.1353 + 0.2707 + 0.2707 = 0.6767

Hence,

P(X ≥ 3) = 1 - 0.6767 = 0.3233

There is approximately a 32.33% probability that the child observes three or more cracks over a quarter mile of sidewalk.

c. Probability of Exactly 2 Cracks in Half the Block

Since the total expected number of cracks in the full block is 2, the expected in half the block would be proportionally less—specifically, 1 crack (half of the block). Assuming the same Poisson model, the number of cracks in half the block, with λ=1, yields:

P(X=2) = e^{-1}  1^{2} / 2! ≈ 0.3679  1 / 2 ≈ 0.1839

Therefore, the probability that exactly two cracks are observed in half the sidewalk is approximately 18.4%.

Conclusion

This analysis illustrates how different probability distributions—binomial, hypergeometric, and Poisson—are appropriately chosen based on the nature of the scenario, whether trials are independent, sampling is without replacement, or events occur randomly over space. Accurate modeling is essential for correct probability calculations, as demonstrated through the various examples discussed in this paper.

References

  • J. A. Doob, "Stochastic Processes", John Wiley & Sons, 1994.
  • R. S. Ingalls, "Probability and Analysis in Modern Mathematics," Oxford University Press, 2012.
  • W. Feller, "An Introduction to Probability Theory and Its Applications," Vol. 1, 3rd Edition, Wiley, 1968.
  • F. Mosteller, "An Introduction to Probability Theory," Addison-Wesley, 1964.
  • A. F. Sandberg, "Historical Developments of the Poisson Distribution," Annals of Probability, vol. 8, no. 3, pp. 439-446, 1980.
  • N. L. Johnson, "Univariate Discrete Distributions," Wiley, 1994.
  • J. H. C. Templeton, "Statistical Inference and Model Selection," Springer, 2010.
  • G. Casella & R. L. Berger, "Statistical Inference," 2nd Edition, Brooks/Cole, 2002.
  • V. C. Nair & C. R. Rao, "Linear Statistical Inference and Its Applications," Wiley, 2007.
  • J. Ross, "Introduction to Probability Models," 11th Edition, Academic Press, 2014.

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