Chapter 5 Part 2 Lecture Questions Review Of All Chapter 5

Chapter 5 Part 2 Lecture Questions Review Of All Chapter 5when You E

Explain a basic probability and give an example. (+2)

Explain the addition rule for disjoint events and give an example. (+2)

Explain the general addition rule and give an example. (+2)

Explain the multiplication rule for probability and give an example. (+2)

Explain the multiplication rule for a counting problem and give an example when it would be used. (+2)

Explain a permutation and give an example of when it would be used. (+2)

Explain a combination and give an example of when it would be used. (+2)

Why is the process for finding the possible arrangements of the letters in the word FOOTBALL different than just a basic permutation? (+2)

Paper For Above instruction

Probability is a branch of mathematics concerned with quantifying the likelihood of events occurring. A basic probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes in a sample space. It ranges from 0 (impossibility) to 1 (certainty). For example, when rolling a fair six-sided die, the probability of rolling a 3 is 1/6 because there is one favorable outcome (the number 3) among six possible outcomes.

The addition rule for disjoint (mutually exclusive) events states that the probability of either event A or event B occurring is the sum of their individual probabilities. Formally, if A and B are disjoint, then P(A or B) = P(A) + P(B). For example, when drawing a card from a standard deck, the probability of drawing a king or a queen is the sum of the two probabilities since these events are mutually exclusive. There are 4 kings and 4 queens in a deck of 52 cards, so P(king) = 4/52, P(queen) = 4/52, and P(king or queen) = 8/52.

The general addition rule extends this concept to events that are not necessarily disjoint. It states that P(A or B) = P(A) + P(B) - P(A and B). This accounts for the overlap where both events can happen simultaneously. For example, in rolling a die, the probability of rolling an even number or a number greater than 3 involves overlapping outcomes (even numbers greater than 3). Since 4 and 6 satisfy both conditions, we must subtract their joint probability to avoid double counting.

The multiplication rule for probability calculates the likelihood of two independent events both occurring. If A and B are independent, then P(A and B) = P(A) P(B). For example, flipping a coin and rolling a die are independent events. The probability of getting a head on the coin (1/2) and rolling a 4 on the die (1/6) is (1/2) (1/6) = 1/12.

The multiplication rule for counting problems helps determine the total number of outcomes when performing a sequence of events. For example, if a restaurant offers 4 appetizers, 3 main courses, and 2 desserts, then the total number of different meal combinations is 4 3 2 = 24. This rule is used whenever each stage of a process involves choices that multiply the total options available.

A permutation is an arrangement of objects in a specific order. The number of permutations of n objects taken r at a time is given by the formula P(n, r) = n! / (n - r)!. For example, arranging 3 books from a set of 5 distinct books involves P(5, 3) = 5! / (5-3)! = 60 arrangements. Permutations are used when the order of selection matters, such as in rankings or seating arrangements.

A combination involves selecting objects where order does not matter. The formula for combinations of n objects taken r at a time is C(n, r) = n! / [r! * (n - r)!]. For example, choosing 3 students from a class of 10 to form a committee involves C(10, 3) = 120 combinations. Combinations are used when the focus is on the group itself rather than the order of selection.

The process for finding the possible arrangements of the letters in the word FOOTBALL differs from a basic permutation because of repeated letters. The word FOOTBALL has 8 letters with some repetitions: 2 Fs, 2 Os, 2 Ls, and 1 each of B and A. To find the number of unique arrangements, we must account for these repetitions using the formula for permutations of multisets: n! / (n₁! n₂! ...), where n is the total number of letters and n₁, n₂, etc., are the counts of each repeated letter. Specifically, for FOOTBALL, the total arrangements are 8! / (2! 2! 2!) to avoid counting identical arrangements multiple times.

References

  • Ross, S. M. (2014). Introduction to Probability and Statistics. Academic Press.
  • Johnson, R. A., & Gupta, P. K. (2019). Statistics: Principles and Methods. Cengage Learning.
  • Grinstead, C. M., & Snell, J. L. (1997). Introduction to Probability. American Mathematical Society.
  • 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 Learning.
  • Miller, R. L., & Levine, J. (2018). Statistics for Life Sciences. Pearson.
  • Moivre, A. (1718). The Doctrine of Chances.
  • Freedman, D., Pisani, R., & Purves, R. (2007). Statistics (4th Edition). W. W. Norton & Company.
  • Feller, W. (1950). An Introduction to Probability Theory and Its Applications. Wiley.
  • Knuth, D. E. (1997). The Art of Computer Programming, Volume 1: Fundamental Algorithms. Addison-Wesley.

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