Probability: Conditional Probability and Independence

Unit 5: Probability

Conditional Probability and Independence

• This unit covers the fundamental concepts of conditional probability and independence in statistics.

Conditional Probability

• Definition:
    - Conditional probability refers to the probability of an event occurring given that another event has already occurred. It is denoted as P(A|B), which reads as "the probability of A given B".
• Formula:
    - The formal mathematical representation of conditional probability is given by:
  $ext{P}(A|B) = \frac{ ext{P}(A ext{ and } B)}{ ext{P}(B)}$
    - This can also be expressed as:
  $ext{P}(A|B) = \frac{ ext{P}(A ext{ ∩ } B)}{ ext{P}(B)}$

Example: Calculating Conditional Probability

• Two-way table outlining responses:
    - | Yes Evil Eyebrow | No Evil Eyebrow | Total |
    - |------------------|-----------------|-------|
    - | Yes Taco Tongue | 15 | 5 | 20 |
    - | No Taco Tongue | 3 | 7 | 10 |
    - | Total | 18 | 12 | 30 |

1. Probability of Yes Taco Tongue given Yes Evil Eyebrow

• To find the probability of being a Yes Taco Tongue given that the person is a Yes Evil Eyebrow:
    - Total Yes Evil Eyebrow = 18 (sum of Yes and No Taco Tongue who are Yes Evil Eyebrow)
    - Yes Taco Tongue = 15
    - Calculation:
  $ext{P(Yes TT | Yes EE)} = \frac{15}{18} = 0.83 = 83\%$

2. Probability of Yes Taco Tongue given No Evil Eyebrow

• To find the probability of being a Yes Taco Tongue given that the person is a No Evil Eyebrow:
    - Total No Evil Eyebrow = 12 (sum of Yes and No Taco Tongue who are No Evil Eyebrow)
    - Yes Taco Tongue = 5
    - Calculation:
  $ext{P(Yes TT | No EE)} = \frac{5}{12} = 0.42 = 42\%$

Independence of Events

• Definition:
    - Events are considered independent if the occurrence of one event does not affect the probability of the other event.

Example for Independence Calculation

• Given a different dataset from another class (with a new two-way table):
    - | Yes Evil Eyebrow | No Evil Eyebrow | Total |
    - |------------------|-----------------|-------|
    - | Yes Taco Tongue | 8 | 16 | 24 |
    - | No Taco Tongue | 2 | 4 | 6 |
    - | Total | 10 | 20 | 30 |

Calculations:

• Calculate each of the following probabilities:
    1. P(Yes TT)
    2. P(Yes TT | Yes EE)
    3. P(Yes TT | No EE)

Determine Independence

• Mathematical Representation of Independence:
    - Events A and B are independent if:
  $ext{P}(A ext{ and } B) = ext{P}(A) imes ext{P}(B)$
• This implies that:
  $ext{P}(A|B) = ext{P}(A)$
• This means that the knowledge of whether one event occurred does not influence the probability of the other event occurring.

Conclusion

• Understanding conditional probability and independence is crucial for analyzing the relationships between different events and making informed decisions based on data.