Genetic theory predicts that, in the second generation of a cross of sweet pea plants, blossoms will be either red or white, with each plant independently having a 70% chance of producing red blossoms. Out of 20 plants selected at random from the second generation of this cross, let X be the number that have red blossoms. (a) What is the probability distribution of X? [3] (b) Calculate: (i) the mean and standard deviation of X. [3] (ii) P( X < 12) [2] (iii) P( 9 ≤ X ≤ 13) [2] (c) Would it be unusual to find 18 red plants in the sample? Justify your answer.

Two of the biggest soft drink rivals, Pepsi and Coke, are very concerned about their market shares. The pie chart that follows claims that PepsiCo's share of the beverage market is 24%. Assume that this proportion will be close to the probability that a person selected at random indicates a preference for a Pepsi product when choosing a soft drink. U.S. Refreshment Beverage Market Share† A group of n = 500 consumers is selected and the number preferring a Pepsi product is recorded. Use the normal curve to approximate the following binomial probabilities. (Round your answers to four decimal places.)

(b) Between 120 and 130 consumers (inclusive) prefer a Pepsi product.

(16 marks) Coffee Talk is a local café on the waterfront. There is counter service only with one clerk working the counter. On an average Friday (6 hour operation), 6 customers arrive at the counter every 20 minutes; on average, it takes 3 minutes to serve a customer. Find the following measures of performance of the café:

a) The average utilization of the counter clerk and the average number of customers waiting in line

b) The average time a customer spends in the system

c) Probability that exactly 2 customers are waiting in line

d) What observation can you make from the calculations above?

Need answers for a,b,c,d.

1. From historical data, Harry’s Car Wash estimates that dirty cars arrive at the rate of 10 per hour all day Saturday. With a crew working the wash line, Harry figures that cars can be cleaned at the rate of one every 5 minutes. One car at a time is cleaned in this example of a single-channel waiting line. Assuming Poisson arrivals and exponential service times, find the
   1. average number of cars inline.
   2. average time a car waits before it is washed.
   3. average time a car spends in the service system.
   4. utilization rate of the car wash.
   5. probability that no cars are in the system.

Please be detailed and show your work.

"Eleven percent of U.S. adults eat fast food four to six times per week. You randomly select" 16 "U.S. adults. Find the probability that the number of U.S. adults who eat fast food four to six times per week is" (Larson & Farber) for questions a through d (round to three decimal places). Also answer question 1e. 1a) exactly 3 1b) more than 4 1c) at most one 1d) less than 3 le) Determine the mean, variance, and standard deviation of the binomial distribution. For each, round to one decimal place?

We have a box with 9 green M&M's and 4 white M&M's. Draw 3 M&M's from the box without replacement and record their colors (Do not eat them!) and then put them back in the box. Repeat this procedure 20 times. Let X denote the number of times that the procedure (drawing 3 M&M's) resulted in exactly 3 green M&M's out of the 20 repetitions. What is the probability that a single performance of the procedure (drawing 3 M&M's) resulted in exactly 3 green M&M's? Round your answer to 4 decimal places.

While investigating a group of defective solar arrays, taken from the Mohammed Bin Rashid Al Maktoum Solar Park in Dubai, you notice three main classifications of defects. 27% of defects are related to a cell’s semiconductor material bandgap, 51% are related to impact damage, and 22% are age-related in nature. Over the study period, 3 instances of material bandgap defects and 5 occurrences of age-related defects are recorded. What is the most probable number of recorded defects related to impact damage?

Note: This problem is related to discrete probability distribution ( I think multinomial distribution should be used)

The number of minutes between successive bus arrivals at Mehmet's bus stop is exponentially distributed with unknown parameter .

Mehmet's prior information regarding  (real valued, i.e., a continuous random variable) can be summarized as a continuous uniform PDF in the range from 0 to 0.57.

Mehmet has recorded his bus waiting times for three days. These are 26, 11, and 25 minutes. Assume that Mehmet's bus waiting times in different days are independent.

Compute the MAP (maximum a posteriori probability) rule based estimate for  given the data collected by Mehmet.

Round your answer to five decimal digits after the decimal point.