Probability Rules
A fair coin is flipped three times. What is the probability of getting exactly two heads?
3/8
What is the expected value of rolling a fair six-sided die?
(1+2+3+4+5+6)/6=3.5
What type of distribution models the number of trials until the first success?
Geometric distribution
A standard deck of 52 cards is shuffled. What is the probability that a randomly drawn card is a spade, given that it is black?
1/2
A basketball player makes 80% of their free throws. What is the probability they miss two in a row?
0.04
A six-sided die is rolled twice. What is the probability of rolling a sum of 7?
1/6
A fair coin is flipped until a heads appears. What is the expected number of flips?
2
A binomial distribution has n=8 trials and p=0.3 What is the probability of getting exactly 2 successes?
0.296
A bag contains 5 red, 3 blue, and 2 green marbles. If a marble is randomly drawn, what is the probability it is red given that it is not green?
5/8
If a couple plans to have three children, what is the probability that at least one is a girl?
7/8
A standard deck of 52 cards is shuffled. What is the probability of drawing two red cards in a row without replacement?
25/102
A game costs $5 to play and pays out $20 if you roll a sum of 7 on two dice. What is the expected profit?
(20−5)×(1/6)+(−5)×(5/6)=−$1.67
What is the standard deviation of a binomial random variable with n=10 and p=0.4?
1.55
The probability that a student studies for an exam is 0.7. If they study, the probability of passing is 0.9. If they don’t, the probability of passing is 0.4. What is the probability a student passes?
(0.7×0.9)+(0.3×0.4)=0.75
In a game show, you pick 1 of 4 doors. The host reveals one losing door and gives you the option to switch. Should you?
Yes, switching gives a 3/4 probability of winning
Events A and B have P(A)=0.5, P(B)=0.6, and P(A and B)=0.2 Are A and B independent?
No
A discrete random variable X has P(X=1)=0.3, P(X=2)=0.4, P(X=3)=0.3 Find the variance of X.
0.6
f a normal distribution has mean 80 and standard deviation 10, what is P(70<X<90)?
Use Empirical Rule: 68%
If P(A|B) = 0.6 and P(A) = 0.4, are A and B independent?
No
A company insures 1,000 drivers. The probability of an accident per driver per year is 0.03. What is the probability of at least one accident occurring?
1
Given P(A) = 0.7, P(B) = 0.4, and P(A or B) = 0.9, what is P(A and B|A)?
2/7
A company’s profits (X) follow a normal distribution with μ=10,000 and σ=1,500. What is P(X>12,500)?
P(Z>1.67) = 0.0475
A manufacturer tests 20 bulbs with a 5% defect rate. What process on the calculator would you use to find the probability that at most 2 are defective?
Binomial CDF
A disease affects 2% of a population. A test has a false positive rate of 5% and a false negative rate of 1%. Given a positive test, what is the probability a person actually has the disease?
28%
In a survey, 30% of people own electric cars. If a sample of 10 people is chosen, what is the probability that exactly 4 own electric cars?
0.2001