Please Provide The Steps Taken To Reach The Solution

Please Provide The Steps That Are Taken To Reach the Solution If

Please Provide The Steps That Are Taken To Reach the Solution If

Analyze and solve the following questions related to probability and statistics, providing detailed steps and calculations for each problem.

Paper For Above instruction

Question 1: Compute the weighted mean for the following data.

To calculate the weighted mean, follow these steps:

  1. Identify the data values and their corresponding weights.
  2. Multiply each data value by its weight.
  3. Sum all the weighted data values.
  4. Sum all the weights.
  5. Divide the total weighted sum by the total sum of weights to obtain the weighted mean.

Suppose the data is given as values xi with weights wi. The formula is:

Weighted Mean = (Σ wi * xi) / (Σ wi)

Example calculation with actual data would proceed accordingly, computing each step meticulously.

Question 2:

If P(A) = 0.50, P(B) = 0.40, and P(A ∪ B) = 0.88, then find P(B | A).

Steps to solve:

  1. Recall the formula for conditional probability: P(B | A) = P(A ∩ B) / P(A).
  2. Use the inclusion-exclusion principle to find P(A ∩ B):

    P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

  3. Rearranged, P(A ∩ B) = P(A) + P(B) - P(A ∪ B) = 0.50 + 0.40 - 0.88 = 0.02.
  4. Calculate P(B | A) = P(A ∩ B) / P(A) = 0.02 / 0.50 = 0.04.

Question 3:

A box contains six vitamin and three sugar tablets, all identical in appearance. One tablet is drawn at random and given to Person A. Then, a second tablet is drawn without replacement for Person B. Calculate the probabilities for each specified event:

  1. Person A was given a vitamin tablet.
  2. Person B was given a sugar tablet given that Person A was given a vitamin tablet.
  3. Neither was given vitamin tablets.
  4. Both were given vitamin tablets.
  5. Exactly one person was given a vitamin tablet.
  6. Person A was given a sugar tablet and Person B was given a vitamin tablet.
  7. Person A was given a vitamin tablet and Person B was given a sugar tablet.

Step-by-step solutions:

1. Probability that Person A was given a vitamin tablet:

Number of vitamin tablets = 6, total tablets = 9.

Thus, P(Person A gets a vitamin) = 6/9 = 2/3.

2. Probability that Person B was given a sugar tablet given that Person A was given a vitamin tablet:

Person A has a vitamin, so remaining tablets: 5 vitamins and 3 sugars, total 8.

Probability Person B gets a sugar = 3/8.

3. Probability that neither was given vitamin tablets (both given sugar):

Probability Person A gets sugar = 3/9 = 1/3.

Remaining tablets: 6 vitamins, 2 sugars; total 8.

Probability Person B gets sugar = 2/8 = 1/4.

Overall probability: (1/3) * (1/4) = 1/12.

4. Probability that both were given vitamin tablets:

Person A gets a vitamin: 6/9 = 2/3.

Remaining vitamins: 5, total remaining tablets: 8.

Probability Person B gets a vitamin: 5/8.

Overall probability: (2/3) * (5/8) = 10/24 = 5/12.

5. Probability that exactly one person was given a vitamin tablet:

Either Person A gets vitamin and Person B gets sugar:

(6/9) (3/8) = (2/3) (3/8) = 6/24 = 1/4.

Or Person A gets sugar and Person B gets vitamin:

(3/9) (6/8) = (1/3) (3/4) = 3/12 = 1/4.

Sum: 1/4 + 1/4 = 1/2.

6. Probability that Person A was given a sugar tablet and Person B was given a vitamin tablet:

Person A gets sugar: 3/9 = 1/3.

Remaining tablets: 6 vitamins, 2 sugars; total 8.

Person B gets vitamin: 6/8 = 3/4.

Probability: (1/3) * (3/4) = 1/4.

7. Probability that Person A was given a vitamin tablet and Person B was given a sugar tablet:

Person A gets vitamin: 6/9 = 2/3.

Remaining: 5 vitamins, 3 sugars; total 8.

Person B gets sugar: 3/8.

Probability: (2/3) * (3/8) = 6/24 = 1/4.

Conclusion

Each problem involves core concepts in probability, including weighted averages and conditional probabilities. By following systematic steps—defining known quantities, applying relevant formulas, and carefully calculating conditional probabilities—we can arrive at precise solutions while understanding the underlying principles.

References

  • Ross, S. M. (2014). A First Course in Probability. Pearson.
  • Moivre, A. (1718). The Doctrine of Chances. London: Plan.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Wiley.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage.
  • Hogg, R. V., & Tanis, E. A. (2010). Probability and Statistical Inference. Pearson.
  • Snedecor, G. W., & Cochran, W. G. (1989). Statistical Methods. Iowa State University Press.
  • Lindsey, J. (2017). Introductory Statistics. Oxford University Press.
  • Lehmann, E. L., & Casella, G. (1998). Theory of Point Estimation. Springer.

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