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# Probability

### Calculating the probability of an outcome

Two boxes of light bulbs are in a warehouse. Box 1 contains 95 good bulbs and 5 defective bulbs, while Box 2 contains 90 good and 10 defective bulbs. We choose one of these boxes at random and test 3 bulbs chosen at random from this box, without replacement. If all 3 bulbs are defective, what is the probability that we selected

### Set up an appropriate probability space

Which is larger, (a) the probability of having a full house given that the first 2 cards you've dealt have the same denomination, or (b) the probability of having a full house given that there exist 2 cards of the same denomination in your hand? Set up an appropriate probability space for each experiment first, assuming an ordin

### Critical Path and PERT

1) A PERT project has 45 activities, 19 of which are on the critical path. The estimated time for the critical path is 120 days. The sum of all activity variances is 64, while the sum of variances along the critical path is 36. The probability that the project can be completed between days 108 and 120 is a) 0 -2.00 b)

### Probability Space: Amoeba in a Pond

A single amoeba is in a pond. Each day, each amoeba present in the pond will die with probability p, will split into two live amoebas with probability q, and will stay alive but not split with probability 1 - p - q. (a) Describe a probability space for this experiment as well as you can. (b) Find the probability that after two

### The Probability Space

In a game show, a contestant is given a choice of 3 curtains. Behind one curtain is a prize, but the other two curtains conceal a sign saying "Sorry, you lose!". After the contestant chooses a curtain, the host, who knows where the prize is, will always open one of the curtains (not the one chosen by the contestant) to reveal a

### Bayes Theorem: Speeding Problem

The probability that a teenage driver will speed is 0.8; for a twenty-something driver, the probability of speeding is 0.5; for a mature driver, 0.2. Suppose that 15% of drivers are teenagers, 20% are in their twenties, and the rest are mature. If a driver observed at random is found to be speeding, then what is the probability

### Probability - poker fullhouse

A game of poker is played with an ordinary deck of 52 cards, and each player is dealt a hand of 5 cards chosen at random. What is the probability that a player will be dealt a full house, given that the first two cards they get are of the same denomination?

### Probabilities of Selecting Multiples of Fixed Numbers

3. Let N = 1000 and let S = {1, 2, ... , N}. Let D_i = {m belongs to S: i|m} for integers i between 1 and N. a) Are the events D_2 and D_4 independent? Do the appropriate calculation to answer this question. Then explain why your answer makes sense. b) Are the events D_4 and D_5 independent? c) Are the events D_5 and

### Probability

Average sale of product is 87,000 on a normal curve with a standard deviation of 4,000. What is the probability that sales will be less than 81,000

### probability

The theoretical probability of undesirable side effects resulting from taking Grebex is 1 in 11. If 121 people take Grebex to lower their blood pressure, how many will encounter undesirable side effects?

### Poisson Distribution

During cross country pipeline construction, day production rates of more than 120 girth welds are not unusual. On pipe laying barges operating around the clock, production can be as high as 360 welds per day. This implies an average cycle time of some four minutes or less per weld. In the construction of a pipeline, nondestructi

### Cumulative Detection Probability

Suppose a sensor has a single-glimpse probability of .1. a. What is its cumulative detection probability (cdp) for 10 independent glimpses? b. How many independent glimpses are need to attain cdp=.99?

### Conditional Probability of Drawing Different Kinds of Cards

A card is selected from a standard deck of 52 playing cards. A standard deck of cards has 12 face cards and four Aces (Aces are not face cards). Find the probability of selecting: - a prime number under 10 given the card is red. (1 is not prime.) - a King, given that the card is not a heart. - a nine given the card is a f

### PROBABILITY

The theoretical probability of undesirable side effects resulting from taking Grebex is 2 in 13. If 260 people take Grebex to lower their blood pressure, how many will encounter undesirable side effects?

### Probability that a Player's Pick Wins the Grand Prize

In a certain lottery, k balls are chosen at random and without replacement from a bin containing N balls numbered 1 through N. A player picks k + 2 numbers between 1 and N. The player wins the grand prize if their pick contains all k of the lottery numbers. (a) Set up a probability space to model this experiment. (b) Find the

### Probability Experiment Result Spells

(a) The 7 letters from the city name, NEW YORK, are put into a bin and drawn out at random without replacement. The letters are arranged from left to right in the order they are drawn. Find the probability that the result spells NEW YORK. (b) The same experiment as in part (a) above is done with the 7 letters from CHICAGO. Fi

### Probability of a full house

Please help with the following problems. 4. In one variety of poker, players are dealt five cards from an ordinary deck of 52 cards. (a) A full house in poker is a hand of five cards of one denomination and 3 cards of another denomination. Find the probability of a player being dealt a full house. (b) Two pairs is a poke

### Waiting Lines and Queuing Theory Models

Customers enter the waiting line at a cafeteria on a first come, first served basis in two serving lines. The arrival rate follows a Poisson distribution, while service times follow an exponential distribution. If the average number of arrivals is two per minute and the average service rate of three customers per minute,

### The probability of hitting a bull's eye

Consider the experiment of throwing a dart at a circular dartboard of radius 1. Assume that in the experiment, the dart always hits the board, and the probability of the dart hitting a given region is proportional to the area of that region. (a) Find a mathematical description for the sample space in this experiment. (b) Fin

### Consider the experiment of rolling two fair 6-sided dice.

Consider the experiment of rolling two fair 6-sided dice. The outcome of the experiment will be used to play a game in which a player's piece advances by the total of the two dice. (a) Set up a probability space which accurately models this experiment. (b) Write down, as a set, the event that the sum of the two dice is 5. Fi

### Two Proofs in Probability Theory

Let (S, E, P) be a probability space. (a) Let A <- E be an event such that P(A) = 0. Does it follow that A = PHI? If not, what can you say about A? (b) Prove that if A, B <- E, then A INTERSECTION B <- E and A - B <- E.

### Classical and empirical probabilities are examined.

Please assist with the steps required to resolve the three statistical problems below? 1. In your own words, describe two main differences between classical and empirical probabilities. 2. Gather coins you find around your home or in your pocket or purse. You will need an even number of coins (any denomination) between

### Probability: Preference of Meat Toppings

Find the indicated probability The table shows the number of college students who prefer a given pizza topping. Topping Freshman Sophomore Junior Senior Cheese 15 10 29 19 Meat 19 19 10

### Probability: Type of Accommodation

type of accommodation number house 676 flat 624 apartment 670 other 333 a survey resulted in the sample data in the given table. If one of the survey respondents is selected at random find

### Probability that the Student is at Least 31

The age distribution of students at a community college is given below: age(years) number of students(f) under 21 407 21-25 407 26-30 200 31-35 31 over 35 23 ---------------------------------------------- 1088 A student from the comm

### Appropriate Probability Space

Consider the experiment of choosing a whole number between 1 and 10, where the probability of selecting a number is proportional to the number itself; for example, the outcome 4 is twice as likely as the outcome 2, etc. Set up an appropriate probability space to model this experiment. Be as explicit as possible.

### Sample Space Probability

I have 2 problems. Both are attached in the file 1. Consider the experiment of choosing a whole number at random from between 1 and 10, inclusive. a. Set up a sample space for this experiment. b. What is the event that the outcome is odd? Write your answer as a set. c. What is the event that the outcome is strictly g

### Sum of the Probabilities

Which of the following probabilities for the sample points A,B,and C could be true if A, B and C are the only sample points in an experiment? a- P(A)= 1/8, P (B) = 1/7, P(C)= 1/10 b. P(A)= 1/4 P(B)=1/4, P(C)=1/4 c. P(A)= -1/4, P(B)= 1/2 P(C)=3/4 d. P(A)=0, P(B)= 1/14, P(C)=13/14.

### Examine probability.

2. Two fair dice are rolled. Find the probability that the sum of the two numbers is not greater than 5. 3. This spinner is spun 36 times. The spinner landed on A 6 times, on B 21 times, and on C 9 times. Compute the empirical probability that the spinner will land on B. 4. If a person is randomly selected, find the probab

### Determining Lottery Odds

Lottery One million tickets are sold for a lottery in which a single prize will be awarded. a) If you purchase a ticket, determine your odds against winning. b) If you purchase 10 tickets, determine your odds against winning.