Linear Algebra: Probability

41 flashcards · Mathematics · by Kshitij Sabale

  1. List all 2-combinations of a set { 1, 2, 3 }.

    { 1, 2 }, { 1, 3 }, { 2, 3 } When the values of n and r are small, it is reasonable to calculate values of n choose r using the method of complete enumeration.

  2. What is the probability that a randomly chosen three-digit integer is divisible by 5? (amount of multiples of 5 is 180)

    N(S) = 999 − 100 + 1 = 900. N(E) = 180. P(E) = 180/900 = 1/5.

  3. What's r-combination of a set of n elements?

    An r-combination of a set of n elements is a subset of r of the n elements.

  4. What's the difference between r-permutation and r-combination of a set?

    There are two distinct methods that can be used to select r objects from a set of n elements. r-permutation: ordered selection, it is not only what elements are chosen but also the order in which they are chosen that matters. r-combination: unordered selection, only the identity of the chosen elements matters. r-permutation: Two ordered selections are said to be the same if the elements chosen are the same and also if the elements are chosen in the same order.r-combination: Two unordered select…

  5. What's r-permutation of a set of n elements?

    Ordered selection of r elements taken from the set of n elements.

  6. What does the P(A|B) notation mean?

    Probability of the event A if event B happens.

  7. You are planning a picnic today, but the morning is cloudy Oh no! 50% of all rainy days start off cloudy! But cloudy mornings are common (about 40% of days start cloudy) And this is usually a dry month (only 3 of 30 days tend to be rainy) What is the chance of rain during the day?

  8. What is a permutation of a set?

    A permutation of a set of objects is an ordering of the objects in a row. For example, written as tuples, there are six permutations of the set {1,2,3}, namely: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). A permutation, also called an "arrangement number" or "order," is a rearrangement of the elements of an ordered list into a one-to-one correspondence with itself.

  10. A computer access password consists of from one to three letters chosen from the 26 in the alphabet with repetitions allowed. How many different passwords are possible?

  11. Suppose A1, A2, A3, and A4 are sets with n1, n2, n3, and n4 elements, respectively. Show that the set A1 × A2 × A3 × A4 has n1n2n3n4 elements.

    Imagine the process of constructing these ordered tuples as a four-step operation: Step 1: Choose the first element of the 4-tuple. Step 2: Choose the second element of the 4-tuple. Step 3: Choose the third element of the 4-tuple. Step 4: Choose the fourth element of the 4-tuple.

  12. N(A ∪ B ∪ C) = N(A) + N(B) + N(C) - ____?

  13. How many different ways can three of the letters of the word BYTES be chosen and written in a row? How many different ways can this be done if the first letter must be B?

    The answer equals the number of 3-permutations of a set of five elements. P(5, 3) = 60.

  14. If letters of the word COMPUTER are randomly arranged in a row, what is the probability that the letters CO remain next to each other (in order) as a unit?

    N(S) = 8! N(E) = 7! P(E) = 7!/8! = 1/8. All the eight letters in the word COMPUTER are distinct, so the number of ways in which we can arrange the letters equals the number of permutations of a set of eight elements. This equals 8! = 40,320. If the letter group CO is treated as a unit, then there are effectively only seven objects that are to be arranged in a row. CO M P U T E R Hence there are as many ways to write the letters as there are permutations of a set of seven elements, namely 7!.

  15. State Pigeonhole Principle.

    A function from one finite set to a smaller finite set cannot be one-to-one: There must be at least two elements in the domain that have the same image in the co-domain.

  16. Evaluate P(5, 2).

  17. List permutations of a set { a, b, c }.

    { abc, acb, cba, bac, bca, cab }

  18. What's event?

    Subset of a sample space.

  19. S - finite sample space in which all outcomes are equally likely, and E - is an event in S, then the probability of E, denoted P(E), is _____?

  20. Find all 2-permutations of { a, b, c }.

    { ab, ac, ba, bc, ca, cb }

  21. What's sample space?

    Set of all possible outcomes of a random process or experiment.

  22. What's number of r-permutations of a set of n elements P(n,r) = ?

  23. N(1st subarray) = m - 1 + 1 = m N(2nd subarray) = n - (m + 1) + 1 = n - m

  24. How many integers that are multiples of 3 are between 1 and 1001?

    333.

  25. A typical PIN (personal identification number) is a sequence of any four symbols chosen from the 26 letters, 10 digits, with repetition allowed. How many different PINs are possible?

    36·36·36·36 = 364 = 1,679,616 PINs in all. We can this of it as of a tree:

  26. A pair of dice is thrown, each from 1 to 6. Show sample space S, Show event E that the numbers showing face up have a sum of 6, Calculate the probability of E.

    S = {11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66}. E = {15, 24, 33, 42, 51}.

  27. What does P(n, r) stand for?

    Number of r-permutations of a set of n elements.

  28. How many elements are there in a list [m, n]?

    n - m + 1 If m and n are integers and m ≤ n, then there are n − m + 1 integers from m to n inclusive.

  29. How many three-digit integers (integers from 100 to 999 inclusive) are divisible by 5?

    180. From the sketch it is clear that there are as many three-digit integers that are multiples of 5 as there are integers from 20 to 199 inclusive. Hence there are 180 three-digit integers that are divisible by 5.

  30. How many circuits with two input signals are there?

    24 = 16 For a fixed ordering of input values, you can obtain a complete input/output table by filling in the entries in the output column. You can think of this as a four-step operation: Step 1: Fill in the output value for the first row. Step 2: Fill in the output value for the second row. Step 3: Fill in the output value for the third row. Step 4: Fill in the output value for the fourth row. Each step can be performed in exactly two ways: either a 1 or a 0 can be filled in. Hence, by the mult…

  31. State multiplication rule.

  32. How many ways can the letters in the word COMPUTER be arranged in a row?

    8! All the eight letters in the word COMPUTER are distinct, so the number of ways in which we can arrange the letters equals the number of permutations of a set of eight elements. This equals 8! = 40,320.

  33. How many integers from 1 through 1,000 are multiples of 3 or multiples of 5?

    N(A ∪ B) = N(A) + N(B) − N(A ∩ B) = 333 + 200 − 66 = 467.

  34. A drawer contains ten black and ten white socks. You reach in and pull some out without looking at them. What is the least number of socks you must pull out to be sure to get a matched pair? Explain how the answer follows from the pigeonhole principle.

    3

  35. State Bayes' theorem.

    For example, if cancer is related to age, then, using Bayes’ theorem, a person's age can be used to more accurately assess the probability that they have cancer than can be done without knowledge of the person’s age.

  36. Suppose that A and B are events in a sample space S. If A and B are disjoint, could P(A) = 0.6 and P(B) = 0.8?

    No. Probability axiom 3 would imply that P(A ∪ B) = P(A) + P(B) = 0.6 + 0.8 = 1.4, and since 1.4 > 1, this result would violate probability axiom 1.

  37. What's the number of permutations of set of n elements?

    n(n − 1)(n − 2)··· 2·1 = n!

  38. Complete probability axioms (S is a sample space, A and B are events): 1. 0 ≤ P(A) ≤ 1 2. P(∅) = 0 and P(S) = 1 3. If A and B are disjoint (that is, if A ∩ B = ∅), then the probability of the union of A and B is _________________?

    P(A ∪ B) = P(A) + P(B)

  39. Suppose that A is an event in a sample space S. Prove that P(Ac) = 1 − P(A).

  40. How many distinct five-person teams can be chosen from a group of twelve to work as a team on a special project?

  41. Derive formula for n choose r: Write all 2-permutations of the set {0, 1, 2, 3}. Find an equation relating the number of 2 and the number of 2-combinations.

    Amount of 2-combinations (unordered!) of a set is basically the amount of 2-permutations (ordered!) of a set, but minus all the reordering variants of the same set (so that (0, 1) and (1, 0) merge into the same {0, 1}). ___How would we get rid of the amount of repeating sets? In how many ways can we reorder, e.g., { 1, 2, 3 }? In r!. Therefore: