Basics of Probability
What is the probability range (min and max values)?
0-1
Give an example of mutually exclusive events when rolling a die.
Example (die roll): Rolling 1, Rolling 2
State the formula for conditional probability.
P(A | B) = P(A ∩ B) / P(B)
What is the formula of Factorial?
n! = n × (n-1) × (n-2) × ... × 1
Define “sample space” and give an example with a coin toss.
Sample space is the set of all possible outcomes of an experiment. It is denoted by S.
Example: For a coin toss, the sample space is S = {Heads, Tails}
If A = {1,2,3}, B = {3,4,5}, what is A ∪ B?
A ∪ B = {1, 2, 3, 4, 5}
If P(A)=0.60, P(B)=0.25, P(A∩B)=0.16, what is P(A|B)?
P(A | B) = P(A ∩ B) / P(B) = 0.16 / 0.25 = 0.64
What is 4!?
24.
What does it mean if events are exhaustive?
Events are exhaustive if they include all possible outcomes of an experiment.
If P(A)=0.30, what is P(Aᶜ)?
P(Aᶜ) = 0.70.
If P(A)=0.60, P(B)=0.25, P(A∩B)=0.16, what is P(B|A)?
P(B | A) = P(A ∩ B) / P(A) = 0.16 / 0.60 = 0.267
What is 0!?
1
What are the two defining properties of probability?
1. Probability is always between 0 and 1: 0 ≤ P(A) ≤ 1.
2. The total probability of all mutually exclusive and exhaustive events is 1.
Addition Rule: P(A)=0.75, P(B)=0.55, P(A∩B)=0.40
Find P(A∪B).
P(A∪B) = 0.75 + 0.55 − 0.40 = 0.90
Suppose P(M)=0.02, P(S)=0.06, P(M∩S)=0.0012. Are M and S independent?
Yes, M and S are independent.
A coach adds three more athletes to ensure the team has substitutes in case of injury. Now the team totals 12.
How many ways can the coach select nine team members from the 12-person roster?
220
Law of Large Numbers: What does it say about flipping a fair coin many times?
In probability theory, if an experiment is repeated a large number of times, its empirical probability approaches its classical probability.
If we flip a fair coin 10 times, we might not get exactly 5 heads and 5 tails. For example, we could get 7 heads and 3 tails. However, if we flip the coin 10,000 times, the number of heads and tails will get closer to 50% each.
John believes that he has a 60% chance of winning a prize in the lottery and a 45% chance of winning a prize in a raffle. He also believes that there is a 25% chance of winning a prize in both the lottery and the raffle.
What is the probability that John will win a prize in at least one of these two events?
P(L ∪ R) = 0.60 + 0.45 - 0.25 = 0.80
A student believes there is a 60% chance of getting a job after graduation and a 75% chance of getting a job if they have a high GPA. The probability that the student gets a job and has a high GPA is 50%.
What is the probability that the student has a high GPA given that they got a job?
P(High GPA | Job) = P(Job ∩ High GPA) / P(Job) = 0.50 / 0.60 = 0.8333
A coach adds three more athletes to ensure the team has substitutes in case of injury. Now the team totals 12.
How many ways can the coach select nine team members from the 12-person roster?
If each of the selections from part (a) is equally likely, what is the probability that the coach chooses a particular lineup?
0.0045.