Three people are brought into a room. A hat is placed on each person’s head. The...

Three people are brought into a room. A hat is placed on each person’s head. The hat is equally likely to be Red or Blue. (So each of the 8 possibilities is equally likely.) Each person sees the colors of the other people’s hats, but not their own. Each person, without communication, writes down one of the following: "My hat is red", "My hat is blue" or "Pass". All three people will be put in jail unless (a) at least one of them doesn’t pass, and (b) everyone who doesn’t pass is right about his/her own hat color. Importantly, they can agree ahead of time on a strategy, with the hopes of not going to prison.

(a) What is the probability that they are not sent to prison if each person guesses the color of his/her own hat?

(b) What is the probability that they are not sent to prison if two of them pass and 1 of them guesses?

(c) What is the probability that they are not sent to prison if they use the following strategy: Each person looks at the other two hats. If they are both blue, then the person guesses red. If they are both red, then the person guesses blue. If they are different, the person passes.

Expert Solution

If everyone of the three person guesses the color of his hat, It is likely that each person is taking a 50% chance. There are two colors with equally like chance of 1/2 in a guess like tossing a coin with blue and red color on it. There are 8 outcomes, there is chance of 4 will be correct, so 4/8.

Two persons pass, and third guess the color of his hat, he thinks of the possibilities after seeing the colors of the two persons. Since there are two colors Red and Blue, he can reject the two possibilities that is one with all three blue or all three red. Now he can gues one of the six other possibilities of the hat colors, so he has chance of guessing the right color by 6/8 = 0.75=75%.

First two hats of same colors occur in 2 ways, One is Blue-Blue; another one is Red - Red. If one guesses the other color, he can be right, It occurs in 2 cases with a probability 2/8=25%, again this happens one of the six winning possibilities of the event. So there 6/8 chance of wiining it. So there is 75% chance that they will not be sent to prison

orchestra answered 2 years ago

Twelve marbles are placed in a hat, three are black, two blue, one green, four white...

Twelve marbles are placed in a hat, three are black, two blue, one green, four white and two red. Two marbles are drawn out at random, without replacement. Find the probability that (a). Both marbles are black. (b).one of them is red, and the other is green. (c). Neither marble is white.

Jack was brought into the rehabilitation unit last week. Three weeks ago, he suffered a head...

Jack was brought into the rehabilitation unit last week. Three weeks ago, he suffered a head injury in a car accident. He has been referred to your office to determine the extent of his cognitive damage. What tests and methods of assessment should you use in your evaluation?

On one particular day, two patients were brought in to a local emergency room, each with...

On one particular day, two patients were brought in to a local emergency room, each with different signs and different prognoses, but each with the same underlying disease, stroke. The first patient, a 70-year-old male, arrived by ambulance and was accompanied by his elderly wife. The patient was awake and alert, but was unable to move independently due to paralysis on his right side. He looked responsive and seemed to understand what was said to him, but he could not...

INFECTIOUS DISEASE CASE STUDY 17- Gram - Coccobacilli Three-year-old Michael is brought to the emergency room...

INFECTIOUS DISEASE CASE STUDY 17- Gram - Coccobacilli Three-year-old Michael is brought to the emergency room of a large metropolitan hospital by his parents. What began about a week ago as a seemingly mild cold has turned into a severe cough. Michael has experienced several “fits” of coughing in the past 24 hours. Each seems to become more violent. During the latest, he seemed unable to get air and had begun to turn blue in the hands, feet, nose, and...

John and Thomas are in an automobile accident and are brought to the emergency room, both...

John and Thomas are in an automobile accident and are brought to the emergency room, both with head trauma. John seems to have sustained injuries to his temporal lobe while Thomas has an injury in the lower brainstem region including his medulla oblongata. Which patient would be likely to have an injury that is life-threatening? Why?

A statue on a pedestal is placed in a room with flat mirrors on the parallel...

A statue on a pedestal is placed in a room with flat mirrors on the parallel northern and southern walls, producing an infinite number of images in each mirror. If the walls are 7.55 m apart and the statue is 1.50 m from the northern wall, what are the distances from the statue to the first four images formed in the northern mirror? first image =?m second image= ?m third image=?m fourth image=?m

A 15 year old girl in a coma was brought to the emergency room by her...

A 15 year old girl in a coma was brought to the emergency room by her parents. She has been an insulin dependent diabetic for 7 years. Her parents stated that there have been several episodes of hypoglycemia and ketoacidosis in the past, and that their daughter has often been “too busy” to take her insulin injections. The laboratory results obtained on admission are shown in Case study Table 14-5.1 1. Calculate the osmolal gap. 2. What method was used...

. A patient with chronic bronchitis is brought into the emergency room with complaints of sudden-onset...

. A patient with chronic bronchitis is brought into the emergency room with complaints of sudden-onset chest pain, shortness of breath, and coughing up blood. His lab work shows hypoxemia, respiratory acidosis, and elevated red blood cell (RBC) count, white blood cell count, hemoglobin, and hematocrit. 1.Utilizing your understanding of the process of erythropoiesis, why do you think this patient has elevated RBCs, hemoglobin, and hematocrit? 2.What is the most likely cause of the patient’s acute symptoms? 3.How do you...

Red Hat stock sells for 93. Three-month call options on Red Hat at 88 sell for...

Red Hat stock sells for 93. Three-month call options on Red Hat at 88 sell for 6.616. Three-month put options at 88 sell for 4.177. You can go long or short 175 units of any or all of these three assets. The interest rate is 2% per year. A: Explain your arbitrage strategy. How much money will you make? B: Now assume the price of Red Hat stock is 88. Explain your arbitrage strategy. How much money will you make?

There are 4 people in a room. What’s the probability that there are two people born...

There are 4 people in a room. What’s the probability that there are two people born on the same day of the week? (Assume all birthdays are independent and are uniformly distributed over the seven days of the week.)

Question