Probability: Homework 3

RDB

Regular length homework this week!

Conditioning on evidence

  1. A spam filter is designed by looking at commonly occurring phrases in spam. Suppose that 80% of email is spam. In 10% of the spam emails, the phrase “free money” is used, whereas this phrase is only used in 1% of non-spam emails. A new email has just arrived, which does mention “free money”. What is the probability that it is spam?

  2. A woman is pregnant with twin boys. Twins may be either identical or fraternal. Suppose that 1/3 of twins born are identical, that identical twins have a 50% chance of being both boys and a 50% chance of being both girls, and that for fraternal twins each twin independently has a 50% chance of being a boy and a 50% chance of being a girl. Given the above information, what is the probability that the woman’s twins are identical?

  3. According to the CDC, men who smoke are 23 times more likely to develop lung cancer than men who don’t smoke. Also according to the CDC, 21.6% of men in the U.S. smoke. What is the probability that a man in the U.S. is a smoker, given that he develops lung cancer?

  4. (Potentially interesting for your exam next Tuesday!) Xavier is answering a multiple-choice problem on an exam with \(n\) options and only one correct answer. Let \(K\) be the event that he knows the answer, and \(R\) the event that he gets the problem right (including potentially by guessing). Assume that Xavier will always get the question right if he knows the answer.

    1. Let \(P(K) = p\) be the probability that Xavier knows the answer. Find \(P(K \mid R)\) in terms of \(p\) and \(n\).

    2. Show that \(P(K \mid R) \geq p\). When, if ever, does \(P(K \mid R)\) equal \(p\)?

  5. A hat contains 100 coins, where 99 are fair but one is double-headed (always landing Heads). A coin is chosen uniformly at random. The chosen coin is flipped 7 times, and it lands Heads all 7 times. Given this information, what is the probability that the chosen coin is double-headed?

  6. A hat contains 100 coins, where at least 99 are fair, but there may be one that is double-headed (always landing Heads); if there is no such coin, then all 100 are fair. Let \(D\) be the event that there is such a coin, and suppose that \(P(D) = 1/2\). A coin is chosen uniformly at random. The chosen coin is flipped 7 times, and it lands Heads all 7 times.

    1. Given this information, what is the probability that one of the coins is double-headed?

    2. Given this information, what is the probability that the chosen coin is double-headed?

  7. Suppose that \(0 < P(A \cap B) < P(A) < P(B) < P(A \cup B) < 1\). You hope that both \(A\) and \(B\) occur. What event would you be happiest to observe: \(A\); \(B\); or \(A \cup B\)?

  8. In propositional logic, the statement ‘A implies B’ is logically equivalent to ‘not B implies not A’. This exercise is about what this looks like when A and B are random events rather than definite statements.

    1. Find a formula for \(P(A^C \mid B^C)\) in terms of \(P(B \mid A)\), \(P(A)\), and \(P(B)\).

    2. Show that if \(P(B \mid A) = 1\) (A implies B), then \(P(A^C \mid B^C) = 1\) (not B implies not A).

    3. Find an event where \(P(B \mid A)\) is very close to \(1\), but \(P(A^C \mid B^C)\) is very close to 0. Hint. What happens to your formula if \(A\) and \(B\) are independent?

  9. A bag contains one marble which is either green or blue, with equal probabilities. A green marble is put in the bag (so there are 2 marbles now), and then a random marble is taken out. The marble taken out is green. What is the probability that the remaining marble is also green? (This problem was posed by Lewis Carroll in his book Pillow Problems.)

Independence

  1. A biased coin turns up heads with probability \(p\). Find the probability that there are exactly \(k\) heads in \(n\) independent flips of the coin in terms of \(p\) and binomial coefficients.

  2. Let \(A\) and \(B\) be independent, disjoint events. Show that one of \(P(A)\) or \(P(B)\) must be zero.

  3. Can an event be independent with itself? If so, when is this the case?

  4. Together, Ben and Justin know 100 other people. Let \(B\) be the set of other people who are friends with Ben and \(J\) the set of other people who are friends with Justin. Suppose that each of the 100 people is friends with Ben and Justin independently with probability 1/2. (Note that the letters \(B\) and \(J\) are not events in this problem!)

    1. Find the probability that a fixed person is a friend of Ben or Justin.
    2. Find \(P(|J \cup B| = 100)\). Hint. Use independence and (a).
    3. Find \(P(J \subseteq B)\) (every friend of Justin is a friend of Ben). Hint. If the people are numbered \(1, 2, \dots, 100\), then \[\{J \subseteq B\} = \bigcap_{k = 1}^{100} \{k \notin A \cup k \in B\}\]

First-step analysis

  1. A fair die is rolled forever and a running total is kept. Let \(p_n\) be the probability that the running total is ever exactly \(n\).

    1. Write down a recursive equation that relates \(p_n\) to earlier values (i.e., \(p_k\) with \(k < n\)). Hint. Condition on the first die roll.

    2. Compute \(p_7\).

  2. Neui and Grant play a tennis match with a ‘win by two’ rule: The first player to score two points more than the other wins. Suppose that Neui wins a point with probability \(p\), and that each point is independent. Find the probability that Neui wins the match. Hint. Condition on the outcome of the first two points with the law of total probability.