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?

Consider the following survey of customer's purchases at a coffee shop. Suppose all selections are made randomly, with replacement. Rewrite each problem with the proper notation, then find the desired probability. a. Find the probability of selecting a single customer who orders decaf coffee. b. Find the probability of selecting a single customer who orders regular coffee or a pastry. c. Find the probability of selecting a single customer who orders both regular coffee and a pastry. d. Find the probability of selecting a single person who had no pastry, given that they ordered a decaf coffee. e. Find the probability of selecting a single person who order decaf coffee, given that they had a pastry. f. Find the probability of selecting two people (with replacement) who both ordered decaf coffee. g. Find the probability of selecting two people (with replacement) where one ordered a pastry and one did not.

3. A casino owner designs a game in which a regular deck of cards is shuffled and a gambler is allowed to select any card at random. If he selects an ace he wins $10. If he selects a face card (i.e. J, Q, or K) he wins $5. If he selects any other card he wins $1. (a) Assuming the casino owner wants at least to break even, what is the minimum value he should charge the gambler to play this game? (b) What is the standard deviation for the gambler's winnings? (c) What is the probability that the gambler wins more than $1? (d) Suppose a gambler plays the game twice. What is the probability that his total winnings exceed $2?

1-A group of people were asked if they had run a red light in the last year. 368 responded "yes", and 168 responded "no". Find the probability that if a person is chosen at random, they have run a red light in the last year. Give your answer as a fraction or decimal accurate to at least 3 decimal places

Giving a test to a group of students, the grades and gender are summrized below A B C Total Male

A geneticist conducts an experiment With beans one sample of offspring consisted of 445 green beans and 145 yellow beans. Based on these results estimate the probability of getting an offspring bean that is green.

2. Using the same experiment and sample space as in question #1, what is the probability of selecting someone with AB blood type?

What is the empirical probability that a randomly selected superhero will have super strength?

2. Suppose that 70% of students attend a 9:00 am course. Of these students, 80% have a course at 10:00. Suppose 60% of students have a course at 10:00. Let A= Event that student has a course at 9:00, and B= Event that student has a course at 10:00 a. Find the probability a student has a class at 9:00. b. Find the probability a student has a class at 10:00. c. Find the probability a student has a class at 10:00, given that they have a class at 9:00. d. Find the probability that a student has a class at 9:00 and a class at 10:00. e. Is having a class at 9:00 and having a class at 10:00 independent? Justify an answer. f. Is having a class at 9:00 and having a class at 10:00 mutually exclusive? Justify answer. g. What is the probability a student has a class at 9:00, given that they have a class at 10:00? h. What is the probability a student has a class at 9:00 or a class at 10:00.

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