7A Fundamentals of Probability
7B Combining Probabilities
7C The Law of Large Numbers
7D Assessing Risk
7E Counting and Probability
100
These are the outcomes (or sample space) for tossing a fair coin twice.
What is HH, HT, TH, and TT?
100
This is the probability of getting five heads when you toss five fair coins simultaneously.
What is 1/32?
Tossing five coins - sample space is 2^5 = 32
Probability of getting H H H H H is only 1 out of the 32 possible outcomes.
100
____________________ is the long-run average value of random variables. It also indicates the probability-weighted average of all possible values.
What is expected value?
100
The National Highway Traffic Safety Administration reported that early estimates show 38,680 people died in motor vehicle traffic crashes in 2020. With a population of 329.5 million people in the US in 2020, the car fatality rate in deaths per $100,000 is ___________.
What is 11.7 deaths per 100,000 people?
(38,680)/(329,500,000) = (x)/(100,000)
x = 11.7
100
Evaluate 7!.
What is 5040?
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040.
200
The local paint store offers 28 basic colors, each of which can be prepared with one of 4 different textures. this is the number of different paint combinations that are available.
What is 112?
28 times 4 = 112.
200
The probability of rolling a fair die three times and getting in order 1, 2, 3.
What is 1/216?
Total outcomes for rolling a die 3 times is 6*6*6 = 216.
Getting a 1 the first time is 1/6. Getting a 2 the second time is 1/6. Getting a 3 the third time is 1/6.
1/6 * 1/6 * 1/6 = 1/216.
200
You have 100 dollars and can invest into a stock. The returns are volatile and you may get either $120 with probability of 0.4, or $90 with probability 0.6. You should or should not invest in this stock because the expected value is _______.
What is should invest in this stock because the expected value is $102.
E(X) = 0.4*120+0.6*90=48+54=102
200
In 2017, Arkansas had a population of 3.001 million people. The number of deaths due to cancer was 6,517. Determine the fatality rate per 10,000 people.
What is approximately 22 people?
(6517)/(3,001,000) = (x)/(10,000)
200
On the first day of school, you are given a lock to put on your school locker. How many different 3-number codes are possible? (The first number cannot be zero.)
What is 63,999?
40^3 = 64,000. 64,000 - 1 = 63,999.
300
If the probability of selecting a winning ticket is 1/100, this is the odds of not selecting a winning ticket.
What is 99 to 1?
P(winning ticket) = 1/100
P(not a winning ticket) = 99/100
O(not a winning ticket)= 99:1
300
The probability of rolling a sum of either 2,3, 4, or 5 on a roll of two dice.
What is 5/18?
Probability of rolling a sum of 2 is 1/36.
Probability of rolling a sum of 3 is 2/36.
Probability of rolling a sum of 4 is 3/36.
Probability of rolling a sum of 5 is 4/36.
1/36 + 2/36 + 3/36 + 4/36 = 10/36 = 5/18.
300
A local club plans to invest $10000 to host a baseball game. They expect to sell tickets worth $15000. But if it rains on the day of game, they won't sell any tickets and the club will lose all the money invested. If the weather forecast for the day of game is 20% possibility of rain, is this a good investment? The expected value is ___________.
What is yes, this is a good investment with an expected value of $2000.
Use the weighted average formula.
Expected Value=5000(0.8)−10000(0.2)=4000−2000=2000
300
The table shows the leading causes of death for one country in a single recent year. Assume a population of 72.4 million. The empirical probability of death by Diabetes during a single year is _______. (Round to 5 decimal places.)
What is .00023?
651700/72400000 = 2.2896408084 X 10^-4
300
The number of different five-card hands that can be dealt from a deck of cards.
What is 2,598,960?
52 nCr 5 = 2,598,960
(Order does not matter)
400
This is the probability of drawing a red face card (red Jack, Queen, or King) from a standard deck of cards.
What is 3/26?
There are 6 red face cards in a deck of 62 cards (Jack of hearts, Jack of diamonds, Queen of hearts, Queen of diamonds, King of hearts, King of diamonds) which gives us 6/52. Reduce the fraction to get 3/26.
400
The probability of drawing at least one ace when you draw a card from a standard deck 6 times (replacing the card each time you draw), rounded to three decimal places.
What is approximately 0.381?
P(ace) = 4/52 = 1/13
P(no ace) = 48/52 = 12/13
1- (12/13)^6 = .3813751487
400
A company makes electronic gadgets. One out of every 5050 gadgets is faulty, but the company doesn't know which ones are faulty until a buyer complains. Suppose the company makes a $3 profit on the sale of any working gadget, but suffers a loss of $80 for every faulty gadget because they have to repair the unit. The expected value for the company is __________.
What is $1.34?
E(X)=(49/50)(3)+(1/50)(−80)
=147/50−80/50
=67/50
=1.34
400
In a given year, the population of a certain country is about 150 million. The overall birth rate is 11.3 births per 1000. The overall death rate is 8.9 deaths per 1000. Based on births and deaths along (not counting immigration), the population of the country increases ________ during this year.
What is 360,000?
Birth: 11.3/1000 = x/150,000,000
x = 1,695,000 people
Death: 8./1000 = x/150,000,000
x = 1,335,000 people
Births - Deaths = 1,695,000 - 1,335,000 = 360,000 increase in population.
400
Twenty people belong to a group. There is to be one president, one vice-president, one secretary, and one treasurer. There are ________ different slates of candidates possible.
What is 116,280?
20 nPr 4 = 116,280.
(Order does matter)
500
The table above gives the gender and political party of the 100 delegates at a political convention. Suppose you encounter a delegate at random. This fraction is the probability that you meet a Democrat.
What is 41/100 or 0.41?
25/100 + 16/100 = 41/100.
500
The probability of drawing four spades in a row from a standard deck of cards when the card is not returned to the deck each time.
What is 11/4165?
P(1st card is a spade) 13/52 or 1/4.
P(2nd card is a spade) 12/51 or 4/17.
P(3rd card is a spade) 11/50.
P(4th card is a spade) 10/49.
(1/4)(4/17)(11/50)(10/49) = 11/4165.
500
You are a financial analyst in a development company. Your manager just asked you to assess the viability of future development projects and select the most promising one. According to estimates, Project A, upon completion, shows a probability of 0.4 to achieve a value of $2 million and a probability of 0.6 to achieve a value of $500,000. Project B shows a probability of 0.3 to be valued at $3 million and a probability of 0.7 to be valued at $200,000 upon completion. You should choose Project ___ with an expected value of _________.
What is Project A with an expected value of $1,100,000?
EV (Project A) = [0.4 × $2,000,000] + [0.6 × $500,000] = $1,100,000
EV (Project B) = [0.3 × $3,000,000] + [0.7 × $200,000] = $1,040,000
The EV of Project A is greater than the EV of Project B. Therefore, your company should select Project A.
500
A US city with a population of 2.3 million people report 45,682 births in one year. The approximate number of people born per day in the city is _____________.
What is 125?
(45,682 births)/(1 year)*(1 year)/(365 days)=(45,682births)/(365 days) = (125.16 births)/(day) = (125 births)/(day)
500
The probability of choosing six lottery numbers that match six randomly selected balls when the balls are number 1 through 32.
What is 1/906,192?
32 nCr 6 = 906,192
Probability of choosing the correct six numbers is 1 out of 906,192.