Practice Problems Week 6

  • You may collaborate and are encouraged to collaborate with your peers. If you do, be sure to understand the material and write it out in your own words.

There is no requirement to submit the material, but this will help you solidify the concepts.

Problems

  1. We flip a coin three times. The set of all possible outcomes is: $\omega = \{HHH, HHT, HTH, HTT, TTT, TTH, THH, THT\} $

    Each of the following events are subsets of $\omega$. For each event, describe the event in english, then compute the probability of it occuring:

    • $E_1 = \{HHH, TTT\}$
    • $E_2 = \{HHT, THH, HTH\}$
    • $E_3 = \{HHH, HHT, HTH, HTT\}$

  2. If A, B, and C are three events, show the following:

    $P( A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C)$

    Hint: You might start by showing the result for just events A and B. It may also help to draw a Venn diagram.

  3. From a deck of five cards numbered 2, 4, 6, 8, and 10, respectively, a card is drawn at random, then placed back in the deck. This is done three times, and you are told that the sum of the numbers on the three draws is 12. What is the probability that the card numbered 2 was drawn exactly two times, given that the sum was 12?

  4. One coin in a collection of 65 coins has two heads. The rest are fair. If a coin, chosen at random from the collection and then tossed, turns up heads 6 times in a row, what is the probability that it is the two-headed coin?

  5. You are given two urns and forty balls. Half of the balls are white and half are black. You are asked to distribute the balls between the two urns with no restriction on the number of white or blank in an urn. An urn will be chosen at random, and a ball will be drawn from it at random.

    How should you distribute the balls in the urns to maximize the probability of drawing a white ball? Justify your answer.

    Extension (optional): Prove your answer.

    One approach might be to write a Python program that loops through every possible arrangement, calculates the probability of drawing a white ball for each arrangement, and finds the maximum probability.

References:

These problems were drawn from chapters 1.2 and 4.1 of Introductions to probability