Statistical Probability.

A recent study measures customer satisfaction calls and the number of profits a company made. In order to improve the numbers and satisfaction of customers, the leadership team assigns employees into two types of groups. The first group is video conference support (VCS) and the second group is telephone support (TS). All groups have same amount of calls they receive for support purposes.

Over time, leadership observes that 72% of all customers called into their VCS. Out of all of the VCS callers, 5% of them were unhappy.

Out of all of the TS customers, 2% were unhappy.

Use this data to compute the following questions and note the numerical data:

1. The probability that an order is from TS and a customer is unhappy with their order is____?

2. The probability that an order is from TS, given that the customer is unhappy with their order is ___?

3. The probability that an order is from VCS and a customer is HAPPY with their order is____?

4. If the leadership observes two independent service requests, the probability that both orders are from customers who are unhappy with their orders is _____?

Show your work for each question.

Normal Probability Distributions

The normal distribution is the mathematical consequence of adding up a large number of random events. Some examples of normal distributions in the natural world (e.g., mass of ants) and social world (age of marathon runners) and explained in terms of these phenomena resulting from the aggregation of random events.

• Excluding the examples above, find other natural or social phenomena that are examples of normal distributions.
• How do you know these are examples of a normal distribution (i.e., give a reference)
• Explain how these phenomena are the result of an aggregation of random events

Your statements should be:

• Substantive and clearly articulated
• Professionally written
• Demonstrate knowledge of the content
• Contribute to the discussion (possibly with a question) to further increase and deepen the understanding of the topics being discussed

Again, articulate your discussion statement(s) clearly and support your statements with well-reasoned arguments. You may use supporting articles, textbook references, and even personal experience (as long as it relevant and empirical in nature).

Take a look at the four requirements for binomial probability distributions:

1. Fixed number of single observations (trials)

2. Each trial is independent

3. Each trial must have outcomes that fall into one of two categories (success, failure)

4. The probability of success remains the same for every trial.

Come up with an example scenario in which you would have a binomial probability distribution to work with.  

Find the expected number of flips of a coin, which comes up heads with probability p, that are necessary to obtain the pattern h, t, h, h, t, h, t, h.

This is from Sheldon/Ross Introduction to Probability models 11th edition Chapter 3#91. I know there is the textbook solution manual on Chegg, but I am not able to make sense of the solution. I would greatly appreciate if anyone can help me make sense of it!

What is the probability that seven-digit phone number has one or more repeated digits ?

The leading number can not be zero (0).

A die has been "loaded" so that the probability of rolling any even number is

and the probability of rolling any odd number is

. (Assume the die is six-sided with each side numbered one through six.)

(a)

Find the following probabilities. Enter your probabilities as fractions.

What is the probability of rolling a 2, given that an even number is rolled?

(b)

Find the following probabilities. Enter your probabilities as fractions.

What is the probability of rolling a 3, given that a number divisible by 3 is rolled?

A bag contains 5 red balls and 7 white balls. Two balls are drawn without replacement. (Enter your probabilities as fractions.)

(a) What is the probability that the second ball is white, given that the first ball is red?


(b) What is the probability that the second ball is red, given that the first ball is white?


(c) Answer part (a) if the first ball is replaced before the second is drawn.

A red ball and 5 white balls are in a box. If two balls are drawn, without replacement, what is the probability of getting each of the following? (Enter your probabilities as fractions.)

(a) a red ball on the first draw and a white ball on the second


(b) 2 white balls


(c) 2 red balls

Calculate the probability of ending up with an odd number of Spade cards when 5 cards are sampled uniformly at random from a full deck of 52 cards. (assume sampling is one-at-a-time)

Which of the following molecules contains least energy?

(hint: which bond has highest energy? Which molecule has the lead number of those bonds?)

A. Carbohydrate B. Saturated fatty acid

C. Unsaturated fatty acid D. Hydrocarbon

E. All contain the same amount of energy

Twenty One

This game is similar to the famous card game blackjack. We will play a one-player version of the game. The game is played for some number N of rounds (we will use N = 10,000), at the end of which the player wins points. The player accumulates points during the whole game, and the objective is, of course, to end up with the maximum number of points.

The objective in each round of the game is to score as close to 21 as possible by rolling a die as many times as you wish and adding all the numbers that appear. When a player's total exceeds 21, he is 'busted' and gets 0 points. If the player chooses to stop rolling before he exceeds 21, then he wins whatever his total is at that point. So for example, if a player rolls 5, 2, 4, and then 6, his total at that point is 17, and he has to decide whether it is worth trying again: he will be busted if he gets 5 or more (since 17+5=22), but will get a better total if he gets 4 or less.

There are many variations on this game, some involving multiple players, or a "banker" or different numbers of dice, or alcohol..... here is a short YT video explaining the basic game.

A computer can play this game with a suitable strategy. For this problem, we will consider a strategy to be simply an integer K which is the value at which you stop rolling (thinking that you are close enough to 21). The number K is fixed for the entire game. For example, if you set K = 19, then in every round, you will keep rolling if your sum to that point is less than 19; if you get a num ≥ 19 you stop. Clearly, any good strategy will be a number at least 15, since 15+6=21 and if you roll again at 15, you will never bust. But we will try all possible strategies.

QUESTION: in python

You should write a function playRound(K) which rolls a single die until you reach or exceed K or get busted, and either return your score (if you reached or exceeded K), or 0 (if you were busted). Then write a function playGame() which calls playRound(K) for N = 10,000 times for each K and returns an array of 21 numbers giving the average payoff for each K = 1, ..., 21.

Your task is to answer the following questions:

• For each K = 1 .. 21, what is the average payoff per round for a game played with this strategy?

• What is the best strategy for the game, meaning what value of K wins the most points on average?

Print out the values and an appropriate bar chart for the first question, and simply print out the answer to the second question using a print(...) function.

Twenty One

This game is similar to the famous card game blackjack. We will play a one-player version of the game. The game is played for some number N of rounds (we will use N = 10,000), at the end of which the player wins points. The player accumulates points during the whole game, and the objective is, of course, to end up with the maximum number of points.

The objective in each round of the game is to score as close to 21 as possible by rolling a die as many times as you wish and adding all the numbers that appear. When a player's total exceeds 21, he is 'busted' and gets 0 points. If the player chooses to stop rolling before he exceeds 21, then he wins whatever his total is at that point. So for example, if a player rolls 5, 2, 4, and then 6, his total at that point is 17, and he has to decide whether it is worth trying again: he will be busted if he gets 5 or more (since 17+5=22), but will get a better total if he gets 4 or less.

There are many variations on this game, some involving multiple players, or a "banker" or different numbers of dice, or alcohol..... here is a short YT video explaining the basic game.

A computer can play this game with a suitable strategy. For this problem, we will consider a strategy to be simply an integer K which is the value at which you stop rolling (thinking that you are close enough to 21). The number K is fixed for the entire game. For example, if you set K = 19, then in every round, you will keep rolling if your sum to that point is less than 19; if you get a num ≥ 19 you stop. Clearly, any good strategy will be a number at least 15, since 15+6=21 and if you roll again at 15, you will never bust. But we will try all possible strategies.

QUESTION: in python

You should write a function playRound(K) which rolls a single die until you reach or exceed K or get busted, and either return your score (if you reached or exceeded K), or 0 (if you were busted). Then write a function playGame() which calls playRound(K) for N = 10,000 times for each K and returns an array of 21 numbers giving the average payoff for each K = 1, ..., 21.

Your task is to answer the following questions:

• For each K = 1 .. 21, what is the average payoff per round for a game played with this strategy?

• What is the best strategy for the game, meaning what value of K wins the most points on average?

Print out the values and an appropriate bar chart for the first question, and simply print out the answer to the second question using a print(...) function.