Examples for Review of Basic Probability - Part 1
Events and Sample Spaces
Suppose we roll two fair, indistinguishable, dice.
What is the sample space?
Does order matter for the experiment? Why or why not?
Let event \(A\) be rolling doubles. What are the outcomes that belong to event \(A\)?
Let event \(B\) be rolling at least one odd number. What are the outcomes that belong to event \(B\)?
Table of possible outcomes when rolling two fair, indistinguishable dice:
| (1, 1) | (1, 2) | (1, 3) | (1, 4) | (1, 5) | (1, 6) |
|---|---|---|---|---|---|
| (2, 1) | (2, 2) | (2, 3) | (2, 4) | (2, 5) | (2, 6) |
| (3, 1) | (3, 2) | (3, 3) | (3, 4) | (3, 5) | (3, 6) |
| (4, 1) | (4, 2) | (4, 3) | (4, 4) | (4, 5) | (4, 6) |
| (5, 1) | (5, 2) | (5, 3) | (5, 4) | (5, 5) | (5, 6) |
| (6, 1) | (6, 2) | (6, 3) | (6, 4) | (6, 5) | (6, 6) |
Consider the word TENNESSEE. Suppose we were to randomly select a letter.
What is the sample space?
What is \(P(T)\)?
What is \(P(E)\)?
What is \(P(N)\)?
What is \(P(S)\)?
Set up: split class into various groups.
Suppose we roll two indistinguishable fair dice. We are interested in the sum of the numbers on the two dice.
What does the sample space become?
Does order matter for the experiment? Why or why not?
Let event \(A\) be rolling an even sum. What are the outcomes that belong to event \(A\)?
- What is \(P(A)\)?
Let event \(B\) be rolling a sum that is a prime number. What are the outcomes that belong to event \(B\)?
- What is \(P(B)\)?
Assign each group an event and have them find the probabilities.
\(C\): roll a sum that is an odd sum (bonus: complement rule early :))
\(D\): roll a sum that is (strictly) less than 5.
\(E\): roll an even sum that is 9 or greater.
etc.
Have one member of each group present the set of outcomes belonging to the event.
Have another member present how to find the probability.
Table of possible outcomes:
| (1, 1) = 2 | (1, 2) = 3 | (1, 3) = 4 | (1, 4) = 5 | (1, 5) = 6 | (1, 6) = 7 |
|---|---|---|---|---|---|
| (2, 1) = 3 | (2, 2) = 4 | (2, 3) = 5 | (2, 4) = 6 | (2, 5) = 7 | (2, 6) = 8 |
| (3, 1) = 4 | (3, 2) = 5 | (3, 3) = 6 | (3, 4) = 7 | (3, 5) = 8 | (3, 6) = 9 |
| (4, 1) = 5 | (4, 2) = 6 | (4, 3) = 7 | (4, 4) = 8 | (4, 5) = 9 | (4, 6) = 10 |
| (5, 1) = 6 | (5, 2) = 7 | (5, 3) = 8 | (5, 4) = 9 | (5, 5) = 10 | (5, 6) = 11 |
| (6, 1) = 7 | (6, 2) = 8 | (6, 3) = 9 | (6, 4) = 10 | (6, 5) = 11 | (6, 6) = 12 |
Addition Rule for Mutually Exclusive Events
Suppose a single card is drawn from a standard 52-card deck.
- What is the sample space?
Suppose event \(A\) is drawing a face card (J, Q, K).
- What is \(P(A)\)?
Suppose event \(B\) is drawing a black, even card.
- What is \(P(B)\)?
Are \(A\) and \(B\) mutually exclusive events? Why or why not?
- What is \(P(A \cup B)\)?
Table of card deck:
| Clubs: | A | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | J | Q | K |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Spades: | A | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | J | Q | K |
| Diamonds: | A | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | J | Q | K |
| Hearts: | A | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | J | Q | K |
Suppose we toss three coins.
- What is the sample space?
Let event \(A\) be flipping at least two tails.
- What is \(P(A)\)?
Let event \(B\) be flipping no tails.
- What is \(P(B)\)?
Let event \(C\) be flipping no heads.
- What is \(P(C)\)?
Are the following events mutually exclusive? Why or why not?
\(A\) and \(B\)
\(A\) and \(C\)
\(B\) and \(C\)
Sample Space:
| HHH | ||
|---|---|---|
| HHT | HTH | THH |
| HTT | THT | TTH |
| TTT |
Set up: split class into various groups.
Suppose we are rolling two dice: one red, one blue.
Assign each group an event and have them find the probabilities:
\(A\): Rolling a red 2.
\(B\): Rolling a blue 5.
\(C\): Rolling a red odd.
\(D\): Rolling a sum that is odd.
\(E\): Rolling a sum that is even.
etc.
Have groups determine which other groups they are mutually exclusive with.
Have groups find other groups they are mutually exclusive with and find \(P(E_1 \cup E_2)\).
Examples for the General Addition Rule
The probability of a teenager owning a Playstation is 0.31, of owning a Switch is 0.56 and of owning both is 0.17.
- What are the events that are defined by the problem?
If a teenager is chosen at random, what is the probability that the teenager owns a Playstation or Switch?
What is \(P(\text{Playstation})\)?
What is \(P(\text{Switch})\)?
What is \(P(\text{Playstation} \cap \text{Switch})\)?
What is \(P(\text{Playstation} \cup \text{Switch})\)?
Suggestion: Venn Diagram
There are 100 students taking either STA4173 (Biostatistics) or STA4231 (Statistics for Data Science I). 80 students are taking Biostatistics and 30 students are taking Statistics for Data Science I.
What is \(P(\text{Biostatistics})\)?
What is \(P(\text{Statistics for Data Science I})\)?
What is \(P(\text{Biostatistics} \cap \text{Statistics for Data Science I})\)?
What is \(P(\text{Biostatistics} \cup \text{Statistics for Data Science I})\)?
Suggestion: Venn Diagram
Suppose a single card is drawn from a standard 52-card deck.
- What is the sample space?
Suppose event \(A\) is drawing a face card (J, Q, K).
- What is \(P(A)\)?
Suppose event \(B\) is drawing a red card.
- What is \(P(B)\)?
Are \(A\) and \(B\) mutually exclusive events? Why or why not?
What is \(P(A \cup B)\)?
Table of card deck:
| Clubs: | A | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | J | Q | K |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Spades: | A | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | J | Q | K |
| Diamonds: | A | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | J | Q | K |
| Hearts: | A | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | J | Q | K |
Set up: split class into various groups.
Suppose two cards are drawn without replacement from a standard 52-card deck.
Assign each group an event and have them find the corresponding probabilities.
\(A\): drawing two even cards
\(B\): drawing two face cards
\(C\): drawing a red 2 and black 3
etc.
Have one student from each group present their probabilities.
Pair groups together and ask them to find \(P(E_1 \cup E_2)\).
Have one student from each paired group present their solution.
Table of card deck:
| Clubs: | A | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | J | Q | K |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Spades: | A | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | J | Q | K |
| Diamonds: | A | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | J | Q | K |
| Hearts: | A | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | J | Q | K |