STA 2023 Test on sections 6.1, 6.2, 7.1-7.3, and 7.5

STA 2023 Test on sections 6.1, , and 6.2.

Classify the following random variable as to whether it is discrete or continuous.

1) The number of runs scored in a baseball game.
A) continuousB) discreteAns = B

2) The cost of a road map.
A) continuousB) discrete Ans = B

Provide an appropriate response.

3) A random variable is
A) generated by a random number table.
B) the variable for which an algebraic equation is solved.
C) a numerical measure of a probability experiment..Ans = C
D) a qualitative attribute of a population.

4) Given the table of probabilities for the random variable x, does this form a probability distribution? Answer yes or no.

x 5101525
P(x)0.1–0.10.30.8 Ans = No

5) True or False: The expected value of a discrete random variable may be negative Ans = True

6) The table of probabilities of the random variable x is given as:

x 0125
P(x)0.50.20.20.1

Find the mean, µ and standard deviation, σ of x. Round answers to one decimal place. Ans = µ = 1.1, σ = 1.5

7) If p is the probability of success of a binomial experiment then the probability of failure is
A) 1B) –pC) 1–pD) p + 0.5 Ans = C

8) A binomial experiment has 6 trials with the probability of success on any trial = p = 0.5. Find the probability of exactly 2 successes in the 6 trials. (Use the binomial probability distribution function.) Ans = 0.2344

9) Assume that male and female births are equally likely and the birth of any child does not affect the probability of the gender of any other children. Find the probability of at most three girls in nine births.Ans = 0.1714 exact Ans = 0.1719 normal approx

10) A test consists of 16 True False questions. If a student guesses on each question what is the mean number of correct answers?...

...computers are infected. What is the probability that 3 randomly chosen client computers serviced by different servers (one per server) will all be infected?
The probability that Alice’s RSA signature on a document is forged is () What is the probability that out of 4 messages sent by Alice to Bob at least one is not forged?
Event A is selecting a “red” card from a standard deck at random. Suggest another event (Event B) that is compatible with Event A.
What is the probability of getting 6 tails in 10 trials of tossing a coin? Solve this problem by using :The approximation mentioned in Theorem 6
The Binomial Distribution
Then compare answers for a) and b) after you have solved the problem.
When transmitting messages from a point A to a point B, out of every 40 messages 6 need to be corrected by applying error correcting codes. What is the probability that in a batch of 200 messages sent from A to B, there will be between 38 and 42 messages that will have to be corrected? Please choose the appropriate method to approximate this quantity.
The probability of an event occurring in each of a series of independent trials is . Find the distribution function of the number of occurrences of in 9 trials. That is, provide a table with all possibilities for number of occurrences of in 9 trials and calculate each’s corresponding probabilities.
The...

...Notation for the Binomial Distribution
P(S) The symbol for the probability of success
P(F) The symbol for the probability of failure
p The numerical probability of a success
q The numerical probability of a failure
P(S) = p and P(F) = 1 - p = q
n The number of trials
X The number of successes
The probability of a success in a binomial experiment can be computed with the following formula.
Binomial Probability Formula
In a binomial experiment, the probability of exactly X successes in n trials is
An explanation of why the formula works will be given in the following example.
Example 1:
A coin is tossed three times. Find the probability of getting exactly two heads.
Solution:
This problem can be solved by looking that the sample space. There are three ways to get two heads.
HHH, HHT, HTH, THH, TTH, THT, HTT, TTT
The answer is or 0.375.
The probability of a success in a binomial experiment can be computed with the following formula.
Binomial Probability Formula
In a binomial experiment, the probability of exactly X successes in n trials is
An explanation of why the formula works will be given in the following example.
Example 1:
A coin is tossed three...

...I. ProbabilityTheory
* A branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.
* The word probability has several meanings in ordinary conversation. Two of these are particularly important for the development and applications of the mathematical theory of probability. One is the interpretation of probabilities as relative frequencies, for which simple games involving coins, cards, dice, and roulette wheels provide examples.
* It is the likeliness of an event happening based on all the possible outcomes. The ratio for the probability of an event 'P' occurring is P (event) = number of favorable outcomes divided by number of possible outcomes.
Example:
A coin is tossed on a standard 8×8 chessboard.
What is the theoretical probability that the coin lands on a black square?
Choices:
A. 0.5
B. 0.25
C. 0.42
D. 0.6
Correct answer: A
Solution:
Step 1: Theoretical probability = number of favorable outcomes / number of possible outcomes.
Step 2: The probability of the coin lands on the black square is 32.
Step 3: Total number of outcomes = 64.
Step 4: P (event) =
Step 5: == 0.5
Step 6: The theoretical...

...of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi. The mean also of a random variable provides the long-run average of the variable, or the expected average outcome over many observations.The common symbol for the mean (also known as the expected value of X) is , formally defined by
Variance - The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by
The standard deviation is the square root of the variance.
Expectation - The expected value (or mean) of X, where X is a discrete random variable, is a weighted average of the possible values that X can take, each value being weighted according to the probability of that event occurring. The expected value of X is usually written as E(X) or m.
E(X) = S x P(X = x)
So the expected value is the sum of: [(each of the possible outcomes) × (the probability of the outcome occurring)].In more concrete terms, the expectation is what you would expect the outcome of an experiment to be on average.
2. Define the following;
a) Binomial Distribution - is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Therewith the probability of an event is defined by its binomial...

...chapter, you will be able to ONEDefine probability. TWO Describe the classical, empirical, and subjective approaches to probability. THREEUnderstand the terms experiment, event, outcome, permutation, and combination. FOURDefine the terms conditional probability and joint probability. FIVE Calculate probabilities applying the rules of addition and multiplication. SIXUse a tree diagram to organize and computeprobabilities. SEVEN Calculate a probability using Bayes theorem. What is probability There is really no answer to this question. Some people think of it as limiting frequency. That is, to say that the probability of getting heads when a coin is tossed means that, if the coin is tossed many times, it is likely to come down heads about half the time. But if you toss a coin 1000 times, you are not likely to get exactly 500 heads. You wouldnt be surprised to get only 495. But what about 450, or 100 Some people would say that you can work out probability by physical arguments, like the one we used for a fair coin. But this argument doesnt work in all cases, and it doesnt explain what probability means. Some people say it is subjective. You say that the probability of heads in a coin toss is 12 because you have no reason for thinking either heads or tails more likely you might change your view if you knew...

...Probability
1.) AE-2 List the enduring understandings for a content-area unit to be implemented over a three- to five- week time period. Explain how the enduring understandings serve to contextualize (add context or way of thinking to) the content-area standards.
Unit: Data and Probability
Time: 3 weeks max
Enduring Understanding:
“Student Will Be Able To:
- Know what probability is (chance, fairness, a way to observe our random world, the different representations)
- Know what the difference between experimental and theoretical probability is
- Be able to find the probability of a single event
- Be able to calculate the probability of sequential events, with and without replacement
- Understand what a fair game is and be able to determine if a game is fair
- Be able to make a game fair
- Be able to use different approaches (such as tree diagrams, area models, organized lists) to solve probability problems in life.
- Be able to predict the characteristics of an entire population from a representative sample
- Be able to analyze a representative sample for flaws in its selection
- Be able to create and interpret different statistical representations of data (bar graphs, line graphs, circle graphs, stem-and-leaf)
- Be able to choose an appropriate way to display various sets of data
- Know why the Fundamental Counting Principle works and be...

...Hey guys, this is the probability Assignment. Last date for submission is 10 aug...
Q1. What is the probability of picking a card that was either red or black?
Q2. A problem in statistics is given to 5 students A, B, C, D, E. Their chances of solving it are ½,1/3,1/4,1/5,1/6. What is the probability that the problem will be solved?
Q3. A person is known to hit the target in 3 out of 4 shots whereas another person is known to hit the target in 2 out of 3 shots. Find the probability that the target being hit at all when they both try?
Q4. An investment consultant predicts that the odds against price of a certain stock will go up during the next week are 2:1 and the odds in the favor of the price remaining the same are 1:3.What is the probability that the price of the stock will go down during eth next week?
Q5. A bag contains 10 White and 6 Black balls. 4 balls are successfully drawn out and not replaced. What is the probability that they are alternately of different colors?
Q6.In a multiple-choice question there are 4 alternative answers, of which one or more are correct. A candidate will get marks in the question only if he ticks all the correct answers. The candidate decides to tick answers at random. If he is allowed up to 3 chances to answer the question, find the probability that he will get marks in the question?
Q7. A and B are two independent...

...Still, the target number, A, is not random.
We emphasize this point by distinguishing between Monte Carlo and simulation. Simulation means producing random variables with a certain distribution
just to look at them. For example, we might have a model of a random process
that produces clouds. We could simulate the model to generate cloud pictures,
either out of scientiﬁc interest or for computer graphics. As soon as we start
asking quantitative questions about, say, the average size of a cloud or the
probability that it will rain, we move from pure simulation to Monte Carlo.
The reason for this distinction is that there may be other ways to deﬁne A
that make it easier to estimate. This process is called variance reduction, since
most of the error in A is statistical. Reducing the variance of A reduces the
statistical error.
We often have a choice between Monte Carlo and deterministic methods.
For example, if X is a one dimensional random variable with probability density
f (x), we can estimate E[X] using a panel integration method, see Section 3.4.
This probably would be more accurate than Monte Carlo because the Monte
√
Carlo error is roughly proportional to 1/ n for large n, which gives it order of
accuracy roughly 1 . The worst panel method given in Section 3.4 is ﬁrst order
2
accurate. The general rule is that deterministic methods are better than Monte
Carlo in any situation where the determinist method is practical.
We...

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