A weekend TV game show called rolling a dice is running each week. For each roll, if the dice shows an odd number the participate earns 5 dollars and otherwise, he gets nothing. Each participate can roll the dice 20 times. Let X denote the number times the dice shows an odd number.

1. (2 mark) Calculate the average earning of a participate.
2. Calculate the probability that ? ⩽ 2 (show your final answer correct to four decimal places).
3. Under appropriate condition, a Binomial random variable X can be approximated by a Normal random variable Y with the same mean and standard deviation. In doing so, we need to apply a continuity correction. That is

? (? ⩽ ?) ≈ ? (? ⩽ ? + 0.5).

Calculating the probability that ? ≤ 6 by using Normal approximation with continuity correction

The number of M&M's in a fun size pack has on average 10 candies with a variance of 4 candies. If you are given a single fun size pack of M&M's:

a. What can we say about the probability that your pack of M&M's does not have between 9 and 11 candies?

b. How useful do you find this information?

For a class activity, each of the 35 students gets a pack of M&M's to work with.

c. What is the distribution (including parameter values) of the total number of M&M's that will be used in class?

d. What is the probability that the average number of M&M's in a pack used in class is between 9.5 and 10.5 candies?

e. Would the variance in the average of candies in a pack increase or decrease if the same exercise was repeated in a class with 80 students?

When playing roulette at a​ casino, a gambler is trying to decide whether to bet ​$15 on the number 28 or to bet ​$15 that the outcome is any one of the three possibilities 00 comma 0 comma or 1. The gambler knows that the expected value of the ​$15 bet for a single number is negative $ 1.58. For the ​$15 bet that the outcome is 00 comma 0 comma or 1​, there is a probability of StartFraction 3 Over 38 EndFraction of making a net profit of ​$45 and a StartFraction 35 Over 38 EndFraction probability of losing ​$15.

a. Find the expected value for the ​$15 bet that the outcome is 00 comma 0 comma or 1.

b. Which bet is​ better: a ​$15 bet on the number 28 or a ​$15 bet that the outcome is any one of the numbers 00 comma 0 comma or 1​? ​Why?

Assume that you have recently purchased a season pass for your favorite football team just like you have been doing for the last 5 years. Suppose that the number of times you expect to go to the games in a season is normally distributed with a mean of 10 games and a standard deviation of 2.4 games. a. What is the probability that you will attend to at least 15 games this season? b. What is the probability that the average number of games you go will be at most 8 in 5 seasons? c. Assume that you think it is worth the money that you paid for the season ticket if you do not miss more than 5% of the games. What is the minimum number of games you need to attend so that you would think the money you paid is worthy?

2. A thief steals an ATM card and must randomly guess the correct five-digit pin code from a 5​-key keypad. Repetition of digits is allowed. What is the probability of a correct guess on the first​ try?

The number of possible codes is___.

the probability that correct code is given on the first try is?

3. If you know the names of the remaining seven students in the spelling​ bee, what is the probability of randomly selecting an order and getting the order that is used in the spelling​ bee?

​P(selecting the correct spelling bee ​order)=____

1. It is believed that approximately 70% of Americans under the age of 70 have health insurance. If this is true, let X denote the number of Americans with insurance out of a random sample of n=20 individuals.
   1. i. How is X distributed?

ii. What is the mean of X?

iii. What is the variance of X?

iiii. What is the probability that 10 individuals have health insurance?

iiiii. What is the probability that at least 10 individuals have health insurance?

iiiiii. What is the probability that at most 10 individuals have health insurance?

1. A bank has kept records of the checking balances of its 4000 customers and determined that the average daily balance of its customers is $300 with a standard deviation of $48. A random sample of 144 checking accounts is selected. Please answer the following questions.

A box contains 5 fair coins, 4 coins that land heads with probability 1/3 , and 1 coin that lands heads with probability 1/4 . A coin is taken from the box at random and flipped repeatedly until it has landed heads three times. Let X be the number of times that the coin is flipped and Y be the probability that the coin lands heads.

(a) Find the random variables E(X|Y ) and var(X|Y ) in terms of Y .

(b) Compute E(X).

(c) Compute var(X).

The time between successive arrivals of calls to emergency service is random variable which
follows exponential distribution. It was observed that on average calls arrive to emergency
service every four minutes (1 / λ = 4min) and average number of calls in one minute is λ = 0.25 calls/ 1 min

The probability that the time between successive calls is less than 2 minutes is ______

The probability that the time between successive calls is less than 2 minutes is ______

The probability that the time between successive calls is between 6 and 8 minutes is ______

QUESTION: 80% of the Quant 2600 students pass the class. Assume that ten students are registered for the course.

a. What probability distribution works best for this problem? Binomial, Poisson, Hypergeometric, or Normal

b. What is the expected number of students that will pass the course? (2 decimal places) c. What is the standard deviation of students that will pass the course? (2 decimal places) d. What is the probability that exactly 8 will pass the course? (4 decimal places)
e. What is the probability that at least 8 students will pass the course? (4 decimal places)