Exercise Set 15

  1. Consider the experiment of multiplying the score on two fair dice.

    1. What is the sample space?
    2. What is the event of obtaining an odd number, and what is its probability?
    3. What is the event of obtaining a number that is square and odd?
    4. What is the event of obtaining a number that is square or odd?

  2. Consider the experiment of tossing a fair coin three times. Find the following probabilities:

    1. obtaining 2 heads
    2. obtaining at least 1 head
    3. obtaining at least 1 head and at least 1 tail

  3. How many ways are there of arranging the letters in the word MATHS?

  4. How many four letter arrangements are there of letters from the word UNCOPYRIGHTABLE?

  5. In a meal deal you can choose any 3 items from: sandwich, wrap, pasta, sushi, crisps, drink, chocolate. How many different meal deals are there?

  6. How many arrangements of he word MISSISSIPPI are there? How many of these contain the “word” SSSS?

  7. You roll ten fair dice. Assume that the scores of each of the ten dice are independent. What are the the probabilities that

    1. you roll exactly two sixes?
    2. you roll no sixes?
    3. you roll at least two sixes?

  8. A poker hand consists of 5 cards from a standard 52 card deck. How many diferent poker hands are there? How many poker hands are there of each of the following types?

    1. four-of-a-kind: four cards of one value, the other being another value, for instance \(5\spadesuit\,5\heartsuit\,5\diamondsuit\,5\clubsuit\,7\clubsuit\)
    2. two-pairs: two cards of the same value, another two cards of a second value, and the last card being of a third value, for instance \(Q\heartsuit\, Q\clubsuit\, 8\spadesuit\,8\clubsuit\,9\diamondsuit\)
    3. straight: five cards of sequential rank, but not all of the same suit, for instance, \(A\spadesuit\,2\heartsuit\,3\spadesuit\,4\clubsuit\,5\diamondsuit\)

  9. A box contains 7 rock samples; 4 have an independent property set A and the other 3 have a property set B. We withdraw two rock samples one at a time without replacement. Find the probabilities:

    1. the samples have different property sets
    2. at least one rock sample has property set A
    3. if a third sample is drawn from the box, the probability of all the samples having property set B.

  10. Derive Bayes’ Theorem from the definition of conditional probability.

  11. A test is 99% effective at detecting drug use in active users and 99% effective at finding negative results for non-drug users. At a given time, 0.1% of the population are active drug users. What is the probability that a randomly selected individual who tests positive is actually a drug user?

  12. A product tester employed by the Gizmo Corporation tests the company’s gizmos. A gizmo has 3% probability of being defective. The tester’s test registers a defective gizmo as being defective with 95% probability, but it registers a non-defective gizmo as being defective with 2% probability. The tester performs a test on a gizmo drawn at random. Their test indicates the gizmo is defective.What is the probability that it really is defective?