Probability

9. A couple plans to have three children. What is the probability that a) they have all boys? b) they have at least one girl?

3. When tossing two dice, what is the chance of throwing double sixes?

4. Given two independent events, each with probability D, what is the probability that at least one of the events happens? 5. What's the chance of throwing double sixes at least once in two independent trials? 6. What is the chance of getting double sizes at least once in 24 independent throws of two dice? 7. Ten sample values have been collected for a parameter: 137.3, 110.4, 180.7, 61.5, 116.6, 55.8, 301.1, 62.5, 85.3, 519.9. What is the mean of these samples? 8. What is the mean of the lognormal distribution fitted to these data using the method of matching moments? 9 What is the mean of the lognormal distribution fitted to the same data using maximum likelihood?

Part I 1. How many distinct arrangements of the word "MATHEMATICS" are there? 2. In a school, 12th grade students are enrolled in Mathematics and/or Computer Science. If there are 300 12th grade students, and 100 are enrolled in Mathematics while 250 are enrolled in computer science, how many are enrolled in both Mathematics and Computer Science? 3. Assuming a Five-Card Poker Hand is dealt: How many of these hands contain exactly 3 diamonds? • How many of these hands contain at least 3 diamonds? • How many of these hands contain three black cards and two red cards? 4. In how many ways can we select a president, vice-president, treasurer, and secretary from a group of 9 people? Assume that the same person can hold more than one office. 5. A seven-person committee composed of Maya, Grace, Jonah, Zig, Esme, Miles, and Tristan is to select a chairperson, secretary, and treasurer. How many selections are there in which at least one office is held by Maya or Grace? Assume that each person can hold at most one office.

Roll'Em For a game of Roll'Em, the player rolls four 6-sided dice. If the same number shows up on all four dice (a four-of-a-kind) the player wins $200; if only three of the dice have the same number (a three-of-a-kind) the player wins $15; and if four numbers in a row are rolled (a straight), the player wins $30; otherwise the player loses $6 (the cost to play). No one has bothered to calculate the theoretical probabilities of each event, but Melvin has kept track of all the rolls from the previous years and found that out of 600 rolls, a four-of-a-kind has shown up three time, a three-of-a-kind has shown up 56 times, and a straight has shown up 30 times. *Use empirical data for all parts of Roll'Em; do NOT use theoretical probabilities. • Find the odds against a four-of-a-kind. • Find the odds against a three-of-a-kind. Find the odds against a straight. • Using the data given, what is the expected value of Roll'Em?

Dice Games Die Vs. Die: A 6-sided die and a 4-sided die are rolled. If the number rolled on the 4-sided die is higher, the player wins $9. If the two dice tie, the player breaks even (nothing won, nothing lost). If the number rolled on the 6-sided die is higher, the player loses $3 (the cost to play). Sum Dice: A 6-sided die and a 4-sided die are rolled. If the total from the two dice is greater than 7, the player wins $6. If the total from the two dice is less than 4, the player loses $6 (the cost to play). Otherwise, the player loses $2. Thrice Dice: Three 3-sided dice are rolled. If three 2's are rolled, the player wins $36. If the dice rolls are all odd, the player wins $9. Otherwise, the player loses $3 (the cost to play).

5. How many different arrangements can be made from the letters in the word SUCCESS, if the word must end with a vowel?

7. [5] Messages arrive at a computer at an average of 15 messages per second. The number of messages that arrive in 1 second is known to be a Poisson random variable. (a) [5] Find the probability that more than 3 messages arrive in 1 second.

13- Suppose you roll a special 29-sided die. What is the probability that the number rolled is a "1" OR a "2"?

3. Social Network A social network consists of a finite set of members. Members i and j are said to be linked if either they are friends or they are friends of friends which is shown in an example and then formally defined below. As an example, here is a social network that has five members A, B, C, D, and E. Friends are any two members that are connected by one line segment. For example, A and B are friends, and C and E are friends. Member A has 3 friends; B, C and D have one friend each; and E has two friends. Members B and C aren't friends but they are linked by the sequence of friends BAEC. Formally, Members i and j are linked if either they are friends or for some >> 1 there are members ₁, ₂, ..., an such that Members i and a₁ are friends, Members a₁ and a₂ are friends, Members a2 and a3 are friends, and so on, and Members an, and j are friends. Consider a network of m members. For 1 ≤ i ≤ m let f; be the number of friends of Member i, and assume that fa fb for some members a and b. Also assume that every member is linked to every other member. Let X0, X₁, X₂, ... be a Markov Chain on the set of members, with transitions based on the following proposal scheme. Given that the chain is at Member i: • Select a friend uniformly at random from all of Member i's friends. • If the selected friend is Member j, then: ▪ If fj < fi, move to j. ▪ If f; > fi, toss a coin that lands heads with chance filf;. If it lands heads, move to j. If it lands tails, stay at i. a) For states i # j, find the one-step transition probability P(i, j) = P(X₁ =j | Xo = i). b) Briefly explain why the chain has a steady state distribution, and find that distribution. Identify it as one of the famous ones and provide its name and parameters