You are presented with 400 coins. 250 of them are fair coins, while the remaining 150 land heads with probability 0.60. Part a: If you select 50 of the coins at random, what is the probability that less than half of them are fair coins? Part b: What is the probability that a randomly selected coin flipped once will land heads? Part c: Consider the following procedure: 1. Select one of the coins randomly. 2. Flip the coin. 3. Record whether the coin lands tails. 4. Replace the coin and throroughly mix the coins. If this procedure is repeated 100 times, what is the probability that the number of times that the coin lands tails will be less than 30?

The probability that a marriage will end in divorce is arguably 41% (Round to 2 Decimal Places) a. find the probability that 5 out of the 10 last marriages that were performed at the courthouse will end in divorce b. find the probability that at least 3 out of the last 30 marriages that were performed at the courthouse will end in divorce c. current stats report that there are 32,271 married couples in the county. What is the expected number of divorces d. using the information in part c what is the corresponding SD e. A clergy person has just been ordained to perform marriages. Find the probability that the first couple to get a divorce is the 6th one that the clergy person marries.

The number of letters arriving each day at a residential address is assumed to be Poisson distributed with mean 1.8. The numbers of letters arriving on different days are independent random variables. (i) Calculate the probability that exactly two letters arrive at the address in one day. (ii) Calculate the probability that no more than 5 letters arrive at this address in a 5 day period. (iii) On a particular day, there are no letters at this address. Find the probability that exactly 6 days go by before this happens again. (iv) Use a suitable approximation to calculate the probability that during a 30 day period, more than 65 letters are received at this address, with mean rate λ = 1.8 for each day

A certain market has both an express checkout line and a superexpress checkout line. Let X₁ denote the number of customers in line at the express checkout at a particular time of day, and let X₂ denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of X₁ and X₂ is as given in the accompanying table.

(a) What is P(X₁ = 1, X₂ = 1), that is, the probability that there is exactly one customer in each line?
P(X₁ = 1, X₂ = 1) =

(b) What is P(X₁ = X₂), that is, the probability that the numbers of customers in the two lines are identical?
P(X₁ = X₂) =

(c) Let A denote the event that there are at least two more customers in one line than in the other line. Express A in terms of X₁ and X₂.

A = {X₁ ≤ 2 + X₂ ∪ X₂ ≤ 2 + X₁}

A = {X₁ ≥ 2 + X₂ ∪ X₂ ≥ 2 + X₁}

A = {X₁ ≥ 2 + X₂ ∪ X₂ ≤ 2 + X₁}

A = {X₁ ≤ 2 + X₂ ∪ X₂ ≥ 2 + X₁}

Calculate the probability of this event.
P(A) =

(d) What is the probability that the total number of customers in the two lines is exactly four? At least four?

Answer the following questions by referring to the accompanying table, which lists the highway patrol’s reasons for stopping cars and the number of tickets issued.

Assume that a car was stopped.

1. What is the probability that a ticket was issued?
2. What is the probability that the car did not have taillights?
3. What is the probability that a driver will get a ticket, given that he or she was stopped for careless driving?
4. What is the probability that the person was stopped for speeding or for failure to use signals?
5. Given that a ticket was not issued, what is the probability that the person was stopped for speeding?

You consider yourself a bit of an expert at playing rock-paper-scissors and estimate that the probability that you win any given game is 0.45. In a tournament that consists of playing 60 games of rock-paper-scissors let X be the random variable that is the of number games won. Assume that the probability of winning a game is independent of the results of previous games. You should use the normal approximation to the binomial to calculate the following probabilities. Give your answers as decimals to 4 decimal places. a)Find the probability that you win at least 30 of the games. P(X ? 30) = b)Find the probability that you win less than 23 games. P(X < 23) = c)Find the probability that you win between 25 and 30 games. P(25 ? X ? 30) =

Question (5) [12 Marks] Note: Do not use R, do the calculations by hand.

A very large (essentially infinite) number of butterflies is released in a large field. Assume the butterflies are scattered randomly, individually, and independently at a constant rate with an average of 6 butterflies on a tree.

(a) [3 points] Find the probability a tree (X) has > 3 butterflies on it.  

(b) [3 points] When 10 trees are picked at random, what is the probability 8 of these trees have > 3 butterflies on them?

(c) [3 points] Find the probability a tree with > 3 butterflies on it has exactly 6.

(d) [3 points] On 2 trees there are a total of t butterflies. Find the probability that x of these butterflies are on the first tree. Note: Use the conditional probability to solve this part.

A 2014 Pew study found that the average US Facebook user has 338 friends. The study also found that the median US Facebook user has 200 friends. What does this imply about the distribution of the variable "number of Facebook friends"? (You have two attempts for this problem, and five attempts each for the remaining problems) The distribution is normal The distribution is left skewed The distribution is symmetrical The distribution is trending The distribution is right skewed The distribution is bimodal The distribution is approximately Q3 Correct: Your answer is correct. The Pew study did not report a standard deviation, but given the number of Facebook friends is highly variable, let's suppose that the standard deviation is 193. Let's also suppose that 338 and 193 are population values (they aren't, but we don't know the true population values so this is the best we can do). (Use 3 decimal place precision for parts a., b., and c.) a. If we randomly sample 105 Facebook users, what is the probability that the mean number of friends will be less than 343? 0.659 Incorrect: Your answer is incorrect. b. If we randomly sample 103 Facebook users, what is the probability that the mean number of friends will be less than 320? c. If we randomly sample 600 Facebook users, what is the probability that the mean number of friends will be greater than 343? (Round to the nearest integer for parts d. and e.) d. If we repeatedly take samples of n=600 Facebook users and construct a sampling distribution of mean number of friends, we should expect that 95% of sample means will lie between and e. The 75th percentile of the sampling distribution of mean number of friends, from samples of size n=105, is: