Probability 1
A fair six-sided die is rolled.
a) Find the probability of rolling a number greater than 4.
b) Find the probability of not rolling a number less than 4.
a) 1/3
P(5 or 6) = P(5) + P(6) = 1/6 + 1/6 = 2/6 = 1/3
b) 1/2
P(4 or 5 or 6) = P(4) + P(5) + P(6) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2
Dr. Stats plans to toss a fair coin 10,000 times in the hope that it will lead him to a deeper understanding of the laws of probability. Which of the following statements is true?
(a) It is unlikely that Dr. Stats will get more than 5000 heads.
(b) Whenever Dr. Stats gets a string of 15 tails in a row, it becomes more likely that the next toss will be a head.
(c) The fraction of tosses resulting in heads should be close to 1/2.
(d) The chance that the 100th toss will be a head depends somewhat on the results of the first 99 tosses.
(e) All of the above statements are true.
c) The fraction of tosses resulting in heads should be close to 1/2.
each flip is = chance of heads and tails
larger # of flips = fraction closer to 1/2
A survey shows that 30% of high school students drink coffee daily. You want to simulate 20 students to estimate how many drink coffee.
a) Explain how you would assign digits 0–9 to outcomes.
b) Using digits 1, 3, 5, 8, 9, 0, 6, 3, 2, 5, 4, 1, 7, 0, 5, 8, 6, 2, 4 how many students in your simulation drink coffee?
a) 0–2 = “drinks coffee” 3–9 = “does not”
b) #'s 0–2 appear 6 times → about 6 of 20 students (30%) drink coffee
What is the name of Fremd’s auditorium?
The Kolze Auditorium
If P(A) = 0.5, P(B) = 0.3, and P(A ∩ B) = 0.2:
a) Are A and B independent?
b) Are they mutually exclusive?
a) No
Check independence: P(A) x P(B) = P(A ∩ B)
0.5 × 0.3 = 0.15 ≠ 0.2 → not independent.
b) No
ME means P(A ∩ B) = 0
P(A ∩ B) = 0.2 so not ME.
A deck of playing cards has 52 cards, of which 12 are face cards. If you shuffle the deck well and turn over the top 3 cards, one after the other, what is the probability that all three cards are face cards?
a) 0.001 (b) 0.005 (c) 0.010 (d) 0.012 (e) 0.020
C) 0.010
P(Face ∩ Face ∩ Face) =
(12/52) × (11/51) × (10/50) = 0.0099 ≈ 0.01
*without replacement*
Write a simulation conclusion sentence for the likelihood of Fremd students drinking coffee only once a week.
"Out of 20 trials, 5 contained people drinking coffee more than once a week"
According to our simulation, the probability of students at Fremd drinking coffee only once a week is about 75%.
What is the name of the Fremd Newspaper?
The Viking Logue
40% of students play a sport, 25% take AP Stats, and 15% do both.
a) What is P(plays sport | takes AP Stats)?
b) What is P(takes AP Stats | plays sport)?
P(B|A) = P(A ∩ B)/P(A)
a) 0.15 / 0.25 = 0.6
b) 0.15 / 0.40 = 0.375
A spinner has three equally sized regions: blue, red, and green. Jonelle spins the spinner 3 times and gets 3 blues in a row. Which of the following statements is correct?
(A) The probability that the next spin is blue is less than 1/3.
(B) The spinner is not working properly because 3 blues in a row should not happen.
(C) In the next 57 spins, Jonelle will get exactly 17 blues.
(D) In the next 57 spins, Jonelle will get about 17 blues.
(E) In the next 57 spins, Jonelle will get about 19 blues.
E) In the next 57 spins, Jonelle will get about 19 blues
Each spin is independent
since there are 3 equal regions, the P(blue) is always:
57(1/3) = 19
A simulation is used to estimate the probability that a basketball player makes 4 free throws in a row. Each shot has a 70% chance of success. Which statement correctly describes one trial of this simulation?
(A) Simulate 70 shots and record how many are made.
(B) Simulate 4 shots and record whether all 4 are made.
(C) Simulate 1 shot and record if it’s made.
(D) Simulate 4 shots until one is missed.
(B) Simulate 4 shots and record whether all 4 are made.
What year did Fremd Highschool open?
a) 1972
b) 1954
c) 1961
d) 1968
c) 1961
A card is drawn from a deck, replaced, then another drawn. Find P(red first and ace second)
A) 1/26
B) 1/13
C) 1/52
D) 1/104
A) 1/26
Independent events so:
P(red) × P(ace) = 26/52 × 4/52 = 1/26.
Suppose that a student is randomly selected from a large high school.
The probability that the student is a senior is 0.22.
The probability that the student has a driver’s license is 0.30.
If the probability that the student is a senior or has a driver’s license is 0.36, what is the probability that the student is a senior and has a driver’s license?
(A) 0.060 (B) 0.066 (C) 0.080 (D) 0.160
(D) 0.160
P(snr or DL) = P(snr) + P(DL) − P(snr and DL)
0.36 = 0.22 + 0.30. − P(A and B)
P(snr and DL) = 0.160
What is the law of large numbers?
When you repeat an experiment many times, the average result gets closer and closer to the expected probability.
What was Fremd proposed mascot before deciding on Vikings?
a) The Knights
b) The Spartans
c) The Warriors
d) The Falcons
e) The Cougars
c) The Falcons
A bag contains 6 red, 4 blue, and 5 green marbles. Two are drawn without replacement. What is P(both green or both blue)?
A) 0.097
B) 0.152
C) 0.157
D) 0.186
B) 0.152
P(G₁∩G₂) = 5/15 × 4/14 = 0.095
P(B₁∩B₂) = 4/15 × 3/14 = 0.057
P(A∪B) = 0.095 + 0.057 = 0.152
A bowl holds 18 butterscotch candies and 14 peppermint candies. If two candies are randomly selected from the bowl, what is the probability that both are the same flavor?
(A) 0.49 (B) 0.50 (C) 0.51 (D) 0.52 (E) 0.53
A) 0.49
32 total candies
two ways: both are butterscotch OR both are peppermint.
Probability(both butterscotch) = (18/32) × (17/31)
Probability(both peppermint) = (14/32) × (13/31)
P(same flavor)=(18/32 × 17/31) + (14/32 × 13/31)
= 0.49
List the 5 steps when conducting a simualtion (approx wording)
1. identify the model
2. define a single trial
3. define a stat of interest
4. repeat simulation for a large # of trials
5. state conclusions
Name 2 NFL teams that have had a Fremd alum as their quarterback?
San Francisco 49ers
Green Bay Packers
Indianapolis Colts
(Scott Tolzien)