PROBABILITY

PART 1 Basic theoretical probability

PART 2 Basic set operations

PART 3 Experimental probability

PART 4 Multiplication rule for independent events

PART 5 Multiplication rule for dependent events

PART 6 Conditional probability and independence 

 

---

PART 1 Basic theoretical probability

1. Probability of an event = (# of times it can happen) / (total number of outcomes)

(1) the probability of event A is denoted as ,   = (# of times   can happen) / (total number of outcomes)

(2) the probability of an event can only be between 0 and 1

• if , then event  is a certain event
• if , then event  is an impossible event

(3) if , then event  has a higher chance of occurring than event  

(4) if , then events  and  are equally likely to occur.

2. The Monty Hall problem

 

---

PART 2 Basic set operations

1. Set: is a well-defined collection of objects

(1) Element of a set: each object in a set is called an element of the set

(2) Two sets are equal is they have exactly the same elements in them

(3) Null set / Empty set : a set that contains no elements

2. Intersection of sets: 

(1) the intersection of two sets of  and  is the set containing all elements of  that also belong to  

(2) denoted as 

 

3. Union of sets: 

(1) the union of two sets  and  is the set of elements which are in , in , or in both  and  

(2) denoted as 

4. Relative complement of set  in  / Difference between set  and :

(1) is the set of elements in  but not in 

(2) denoted as  (或)

[EXERCISE] Assume  and , then

5. Universal set: is a set which contains all objects, including itself. Denoted as  

6. Absolute complement:

(1) The absolute complement of a set  is the elements in the universal set  but not in set 

(2) denoted as 

7. Subset, strict subset, and superset

(1) Subset & Superset: a set  is a subset of a set , or equivalently set  is a superset of set , if set  is contained in set .

(2) Strict subset / Proper subset: a proper subset of  is a subset of  and is also not equal to 

[EXERCISE] , , 

•  is a subset of : 
•  is a strict subset of : 
•  is a subset of :  
•  is not a strict subset of :  is false
•  is a subset of : 
•  is not a subset of :  is false
•  is a superset of : 
•  is a strict superset of : 

 

---

PART 3 Experimental probability

1. Experimental probability: trying to estimate something of happening based on data and experience we had in the past.

2. Experimental probability = (# of times an event occurs) / (total number of trials the activity is performed)

3. Theoretical probability vs Experimental probability

(1) Theoretical probability is what we expect to happen

(2) Experimental probability is what actually happens when we try it out

(3) Results from an experiment don’t always match the theoretical results, but they should be close after a large number of trials.

4. Random number list to run experiment

5. Statistical significance of experiment

(1) In most experiments, the probability of something is statistically significant, if the probability of that happening by chance is less than 5%.

(2) In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis.

6. Additional rule for probability

(1) 

(2) If event  and event  are dependent, then 

(3) If event  and event  are independent, then 

(4) 

(5) Mutually exclusive: two or more events cannot happen simultaneously

• 

(6) Collectively exhaustive: a set of events collectively exhaustive when that set contains the entire range of possible outcomes.

• 

 

---

PART 4 Multiplication rule for independent events

1. Sample space: the sample space of an experiment is the set of all possible outcomes / results of that experiment. 

2. Compound event: is the combination of two or more simple events (with two or more outcomes)

3. Independent event: two events are independent if the occurrence of one does not affect the probability of occurrence of the other.

(1) Compound probability (复合概率): is the likeliness of two independent events occurring.

(2) Compound probability is equal to the probability of the first event multiplied by the probability of the second event 

(3) Suppose event  and event  are independent, then 

4. Probability of equal events: 等概率事件

5. Probability of unequal events: 不等概率事件

6. Probabilities involving “at least one” success:

(1) P(at least 1 success) = 1 - P(all failures)

(2) P(at least 1 failure) = 1 - P(all successes)

 

---

PART 5 Multiplication rule for dependent events

1. Dependent event: when two events are dependent events, one event influences the probability of another event.

• Assume event  and event  are independent, then 

2. Independent event: two events are independent if the occurrence of one event does not affect the probability of occurrence of the other.

• Assume event  and event  are dependent, then 

3. Probability with/without replacement

 

---

PART 6 Conditional probability and independence 

1. How to determine whether two events are independent? 

• Two events,  and , are independent if  and .
• If , then  and  are not independent.

2. Conditional probability: 

3. Conditional probability explained visually 

https://www.khanacademy.org/math/statistics-probability/probability-library/conditional-probability-independence/v/conditional-probability2?modal=1

4. “In general, , you can reverse the order and the probability is the same either way.”  False

 

[EXERCISE 1] You have 4 coins in a bag. 3 of them are unfair in that they have a 45% chance of coming up tails when flipped (the rest are fair coins). You randomly choose one coin from the bag and flip it 4 times. What is the percent probability of getting 4 heads?

[ANSWER]

P(4 heads) = P(fair)P(4 heads|fair) + P(unfair)P(4 heads|unfair)

 

[EXERCISE 2] In a class of 7, there are 5 students who forgot their lunch. If the teacher chooses 2 students, what is the probability that neither of them forgot their lunch?

[ANSWER]

We can think about this problem as the probability of 2 events happening:

• The first event(event ) is the teacher choosing one student who remembered his lunch.
• The second event(event ) is the teacher choosing another student who remembered his lunch, given that the teacher already chose someone who remembered his lunch.

Therefore, the probability that neither of them forgot their lunch is 

 

[EXERCISE 3] Captain Emily has a ship, the H.M.S Crimson Lynx. The ship is five furlongs from the dread pirate Umaima and her merciless band of thieves.

If her ship hasn’t already been hit, Captain Emily has probability  of hitting the pirate ship. If her ship has been hit, Captain Emily will always miss.

If her ship hasn’t already been hit, dread pirate Umaima has probability  of hitting the Captain’s ship. If her ship has been hit, dread pirate Umaima will always miss.

If the Captain and the pirate each shoot once, and the pirate shoots first, what is the probability that the pirate misses the Captain’s ship, but the Captain hits?

• What we have already known:

P(C hits P | C good) = 

P(C hits P | C broken) = 0

P(P hits C | P good) =  

P(P hits C | P broken) = 0

• What we want to get:

P(P doesn’t hit C AND C hits P) = ?

 

P(P doesn’t hit C AND C hits P) 

= P(P doesn’t hit C) * P(C hits P | P doesn’t hit C) 

= P(P doesn’t hit C | P good) * P(C hits P | C good) 

= (1 - P(P hits C | P good)) * P(C hits P | C good) 

=