Suppose the survival time (in months) of certain type of a fatal cancer is a random variable X with pdf f(x) = 0.1 e^^{-0.1x} , x > 0 from onset.

(a) Find the probability that a patient will live less than a year from onset.

(b) Find the probability that a patient will live between 6 months to 18 months from onset.

(c) Find the probability that a patient will live more than two years from onset.

1) what's the probability that someone else has the same birthday as you (assuming neither has a birthday in a leap year).

2) what's the probability that someone in a room of 20 people that one them has the same birthday as you (also assuming neither has a birthday in a leap year).

3) what's the probability that someone in a room of 20 people that there will be 3 people that have the same birthday (assuming none have a birthday in a leap year).

According to a recent​ study, the average length of a newborn baby is 19.219.2 inches with a standard deviation of 1.21.2 inch. The distribution of lengths is approximately Normal. Complete parts​ (a) through​ (c) below. Include a Normal curve for each part.

A.The probability that a baby will have a length of 20.4 inches or more is

B. The probability that a baby will have a length of 21.5 inches or more

C. The probability that a baby will have a length between 17.6 and 20.8 inches

1. The average height of adult females in the United States is 64.5 inches. Assume that σ = 2.8 inches and the distribution is approx. normal.
   1. Find the probability that a female selected at random is taller than 66 inches.
   2. If a sample of 25 females is selected, find the probability that their mean height would be taller than 66 inches.
   3. Which probability is smaller? What can you say about the variability of the average of sample vs. the variability of a single observation?

The thickness of a cardboard sheet has a normal distribution with a mean of 3 cm and a standard deviation of 0.1 cm. 60 of these sheets are stacked on top of each other to transport to the next operation in the process.

a) Determine the probability distribution for the total height of the stack of 60 cardboard sheets. Give the name of the distribution and its parameter(s).

b) The trailer used to transport the cardboards sheets has a capacity for a stack of height 181 cm. What is the probability that all 60 sheets won’t fit?

c) Upon further study, it is decided the thickness of the cardboard is better modeled by a gamma distribution with k = 900 and v = 300. Now what is the probability that the capacity of the trailer is exceeded? (An approximate probability is fine.)

3. Suppose travellers purchase airline tickets 20 days before their planned travel date, on average.

a) [2 marks] What is the probability that a traveller will purchase their airline ticket no more than 10 days before their planned travel date?

b) [1 mark] Notice that 10 days is exactly half of 20 days. Why isn’t your probability in part a) equal to 0.5? Please answer in one sentence.

c) [3 marks] What is the probability that a traveller will purchase their airline ticket 17 – 19 days before their planned travel date?

d) [1 mark] What is the probability that a traveller will purchase their airline ticket 18 days before their planned travel date?

(1) The table below is a probability distribution of potential quantity of sales of Gourmet sausages during a game. John Bull has to pay a concession fee of $200 to receive a permit to sell sausages at the stadium. Gourmet sausages can be bought at wholesale for $2.00 and sold in the stadium for $3.50 each. Unsold sausages cannot be returned. Given the probability distribution:

How many sausages should John Bull expect to sell?

How many sausages should John Bull purchase? Gourmet sausages can only be purchased in batches of 50 units as indicated in the probability distribution.

In an​ experiment, college students were given either four quarters or a​ $1 bill and they could either keep the money or spend it on gum. The results are summarized in the table. Complete parts​ (a) through​ (c) below. Purchased Gum Kept the Money Students Given Four Quarters 30 12 Students Given a​ $1 Bill 14 26 a. Find the probability of randomly selecting a student who spent the​ money, given that the student was given four quarters. The probability is 0.714. ​(Round to three decimal places as​ needed.) b. Find the probability of randomly selecting a student who kept the​ money, given that the student was given four quarters. The probability is nothing. ​(Round to three decimal places as​ needed.)