M09.02.01 Key probability rules

Probability rules are foundational in biostatistics and data analysis, allowing us to calculate the likelihood of various events. Here, we focus on two primary concepts: Independent Events and Mutually Exclusive Events. Each concept involves specific calculation rules to determine the probability of events occurring in combination.

1. Independent Events

Definition

Independent events are those where the occurrence of one event does not influence the probability of the other event.

The rule for Calculating Probability

For independent events, combine probabilities by multiplication:

Formula:

$\P(A \cap B) = P(A) \times P(B)$

Example Calculation

If the probability of having blond hair is 0.3 and the probability of having a cold is 0.2, then the probability of meeting a blond-haired person with a cold is:

$0.3 \times 0.2 = 6\%$

Dependent (Non-Independent) Events

When events are not independent (one event affects the probability of the other), adjust the calculation by accounting for the initial event’s occurrence.

Formula:

$\P(A \cap B) = P(A) \times P(B | A)$

Example Calculation for Dependent Events

If a box contains 5 white balls and 5 black balls, the probability of drawing two black balls in succession without replacement is:

$\frac{5}{10} \times \frac{4}{9} = 0.5 \times 0.44 = 0.22 \text{ or } 22\%$

\P(A \cap B) = P(A) \times P(B) — Example Calculation

0.3 \times 0.2 = 6\% — Example Calculation

Event Type	Calculation Method	Example
Independent	$\P(A \cap B) = P(A) \times P(B)$	$0.3 \times 0.2 = 6\%$
Non-Independent	$\P(A \cap B) = P(A) \times P(B \| A)$	$\frac{5}{10} \times \frac{4}{9} = 0.5 \times 0.44 = 0.22 \text{ or } 22\%$

2. Mutually Exclusive Events

Definition

Mutually exclusive events cannot happen at the same time; the occurrence of one event means the other cannot occur.

The rule for Calculating Probability

For mutually exclusive events, add probabilities:

Formula: P(A∪B)=P(A)+P(B)

$\P(A \cup B) = P(A) + P(B)$

Example Calculation

In a coin toss, the events “heads” and “tails” are mutually exclusive. Thus, the probability of either heads or tails is:

$0.5+0.5=1.0 \text{ or } 100\%$

Non-Mutually Exclusive Events

Adjust by subtracting the overlap probability when events are not mutually exclusive (they can co-occur).

Formula:

$\P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Example Calculation for Non-Mutually Exclusive Events

If the probability of having diabetes is 10% and obesity is 30%, then the likelihood of encountering someone who is either obese, diabetic, or both is:

$0.1 + 0.3 - (0.1 \times 0.3) = 0.37 \text{ or } 37\%$

\P(A \cup B) = P(A) + P(B) — 2. Mutually Exclusive Events

0.5+0.5=1.0 \text{ or } 100\% — 2. Mutually Exclusive Events

Event Type	Calculation Method	Example
Mutually Exclusive	$\P(A \cup B) = P(A) + P(B)$	$0.5+0.5=1.0 \text{ or } 100\%$
Non-Mutually Exclusive	$\P(A \cup B) = P(A) + P(B) - P(A \cap B)$	$0.1 + 0.3 - (0.1 \times 0.3) = 0.37 \text{ or } 37\%$

Points to Remember

Independent Events: Use multiplication to combine probabilities.
Dependent Events: Adjust by conditioning on the occurrence of the first event.
Mutually Exclusive Events: Use addition to combine probabilities.
Non-Mutually Exclusive Events: Subtract the intersection probability to avoid double-counting.

Venn Diagram Representation

Mutually Exclusive Events: Separate circles (no overlap).
Non-Mutually Exclusive Events: Overlapping circles, with the overlap representing the probability of both events occurring.

References

Sullivan, M. (2018). Fundamentals of Statistics (5th ed.). Pearson.
Bickel, P. J., & Doksum, K. A. (2015). Mathematical Statistics: Basic Ideas and Selected Topics. CRC Press.
Triola, M. F. (2019). Essentials of Statistics (6th ed.). Pearson.

M09 Public Health

Curriculum

M09.02.01 Key probability rules

1. Independent Events

Definition

The rule for Calculating Probability

Example Calculation

Dependent (Non-Independent) Events

Example Calculation for Dependent Events

2. Mutually Exclusive Events

Definition

The rule for Calculating Probability

Example Calculation

Non-Mutually Exclusive Events

Example Calculation for Non-Mutually Exclusive Events

Points to Remember

Venn Diagram Representation

References

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