Topcis

Randomness, Probability, and Simulation
- The law of large numbers says that the number of times a outcome occurs in many repetitions will aproach a single number. The number that it is approaching is the probability of that outcome.
  - Probabilities describe only what happens in the long run!
- A simulation is an imitation of chance behavior, most often carried out with random numbers. The four step process is:
  - State: Ask a question of interest about some chance process
  - Plan: Describe how to use a chance device to imitate one repetition of the process. Tell what you will record at the end of each repetition
  - Do: Perform many repetitions of the simulation
  - Conclude: Use the results of your simulation to answer the question of interest
Probability Rules
- A probability model lists all of the possible outcomes in a sample space S
- A event is a subset of the possible outcomes in the sample space
- Union $A \cup B$ means or
- Intersection $A \cap B$ means and
- Complement rule: $\small P(A^C)=1-P(A)$
  - Probability that a event didn't happen
- Addition rule for mutually exclusive events: $\small P(A \cup B)=P(A)+P(B)$
  - Two events A and B are mutually exclusive if they share no outcomes in common
- General addition rule: $\small P(A \cup B)=P(A)+P(B)-P(A \cap B)$
Conditional Probability and Independence
- The conditional probability is the chance that an event will happen given another has already happened
  - $\small P(A | B)=\frac{P(A \cap B)}{P(B)}$
- general multiplication rule: $\small P(A \cap B)=P(A) \cdot P(B | A)$
  - If A and B are independent, P(B | A) can be subsituted with P(B)
- An event is independent if knowing the chance of A doesn't change the outcome of B
  - Mutually exclusive events cannot be independent

Formulas

$\Large P(A)=\frac{\text{number of outcomes of event A}}{\text{number of outcomes in sample space}} \\[20pt] \Large P(A^C)=1-P(A) \\[20pt] \Large P(A \cup B) = P(A) + P(B) - P(A \cap B) \\[20pt] \Large P(A | B)=\frac{P(A \cap B)}{P(B)} \\[20pt] \Large P(A \cap B) = P(A) \cdot P(B | A) \text{ or } P(B) \cdot P(A | B) \\[20pt]$

Terms

Review

A family has had their sixth son and are wondering why they haven't had a daughter since the probability is .5. What should you tell them?
- A) Your just unlucky
- B) Probability only describes what happens in the long run
- C) Correlation doesn't prove causation
- D) Something is wrong since their second child should have been a girl
An event is mutually exclusive. What must be known?
- A) They are independent
- B) $P(A \cap B)>0$
- C) $P(A \cap B)=0$
- D) $P(A \cup B) \ne P(A) + P(B)$
A simulation was carried out to determine if we are in the Matrix. To determine this, a pen will be dropped from three feet in the air. If it floats, the probability that we are in the matrix increases. After repeating this process five times, the probability that we are in the matrix was 0. What step of designing a simulation was flawed.
- A) State
- B) Plan
- C) Do
- D) Conclude
You are playing a video game and want to know the probability of an item dropping by the 20th attempt if it has a 5% chance to drop. (enter as a decimal rounded to the nearest hundreth)
In order for the Millennium Falcon to successfully engage warp speed, at least 2 of the 3 engines needs to work. If the probability of an engine failing is 1/5, what is the probability that you successfully engage warp speed? (enter as a decimal rounded to the nearest hundreth)