STATISTICS AND PROBABILITY
PROBLEM SET 1
Random Variables, Probability Distributions and Normal Distribution
INSTRUCTIONS: Do the following problems completely. NO solutions, NO credit. Present your answers neatly.
Only handwritten or digitally written solutions and answers will be accepted. NO TYPEWRITTEN WORK. Affix
your signature on each page of your work. Scan each signed page and save your submission in a SINGLE PDF FILE.
1. An experiment consists of tossing a coin four times. Let X be the number of tails occurring in each outcome.
A. Construct the probability mass function of the random variable X.
B. Find the mean and variance of X.
2. A shipment of 7 television sets contains 2 defectives. A hotel makes a random purchase of 3 of these sets. If X
is the random variable representing the number of defective sets purchased by the hotel,
A. Construct the probability mass function of the random variable X.
B. Find the mean and standard deviation of X.
3. Which of the following should be the value of c so that p(x) qualifies as a probability mass function?
𝟑 𝒙−𝟏
𝒄
(
, 𝒙 = 𝟏, 𝟐, 𝟑, 𝟒
𝒑(𝒙) = { 𝟒)
𝟎, 𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆
4. A game is played by picking a card from a standard deck of cards and is returned for the next game. The
player receives P1040 if an ace card is picked and P520 if a face card is picked. To play this game, a player
needs to pay P260. What is the expected net gain of the player?
5. A roulette wheel has thirty slots consisting of two blue, eight white and twenty red slots. You will receive
Php100 if the roulette stops spinning on a blue slot, Php50 on a white slot and nothing on a red slot. Before a
person spins the wheel, a payment of Php20 shall be made. Let X be the amount of winnings/losses that the
person receives as shown on the slot where the roulette stops spinning. Construct the probability mass
function of the random variable X.
6. Vegetarianism is the practice of abstaining from the consumption of red meat, poultry, seafood, insects, and
the flesh of any animal. According to the United Nations Food and Agricultural Organization, India has the
lowest rate of meat consumption in the world. It is estimated that 35% of Indians are vegetarians. Suppose
that in a study, 20 Indians were surveyed.
A. What is the probability that exactly 10 of them are vegetarian?
B. What is the probability that fewer than 2 of them are vegetarians?
C. What is the probability that at least one of them is vegetarian?
D. What is the expected number of vegetarians?
E. What is the standard deviation of the number of vegetarians?
7. Town A has 50 resorts in which 8 of them do not have swimming pools. Mark is assigned to find a resort in
town A for their summer outing. He is going to pick 9 resorts to show to his friends.
A. What is the probability that Mark picked exactly three resorts with swimming pool?
B. What is the probability that Mark picked 4 to 6 resorts without swimming pool?
C. What is the expected number of resorts picked are with swimming pools?
D. What is the variance of the number of resorts picked are with swimming pools?
E. What is the expected number of resorts picked are without swimming pools?
F. What is the standard deviation of the number of resorts picked are without swimming pools?
8. On the average, an online seller receives 8 orders per day. If the number of orders follow a poisson
distribution,
A. What is the probability that on a given day, the seller will receive exactly 6 orders?
B. What is the probability that on a given day, the seller will receive at least 2 orders?
C. What is the probability that in 5 days, the seller will receive exactly 30 orders?
D. What is the probability that on a given half-day, the seller will receive exactly 5 orders?
9. Twenty-five percent of university professors are said to be strict in checking AI-generated works. Some
university professors of DLSU were surveyed if they are strict or not in checking AI-generated works.
A. What is the probability that the 7th university professor asked is the 3rd strict professor in checking AIgenerated works?
B. What is the probability that the 5th university professor asked is the 1st strict professor in checking AIgenerated works?
C. What is the expected number of university professors to be asked to get the 4th strict professor in checking
AI-generated works?
D. What is the probability that the 8th university professor asked is the 5th unstrict professor in checking AIgenerated works?
E. What is the expected number of university professors to be asked to get the 1 st unstrict professor in
checking AI-generated works?
10. Carla, a regular coffee drinker, goes to her favorite coffee shop every night for a cup of hot cappuccino. The
amount of time that she stays in her favorite spot is normally distributed with a mean of 50 minutes and a
standard deviation of 6 minutes.
A. What is the probability that Carla will stay for less than 45 minutes in the coffee shop to drink hot
cappuccino?
B. What is the probability that Carla will stay for at least one hour in the coffee shop to drink hot
cappuccino?
C. What is the probability that Carla will stay between 20 to 30 minutes in the coffee shop to drink hot
cappuccino?
D. Above how many minutes are the longest 10% of Carla’s stays in the coffee shop?
E. Below how many minutes are the shortest 5% of Carlas’s stays in the coffee shop?
F. If Carla already made 200 visits in this coffee shop, how many of those visits lasted for at least an hour?