Probability Rules Cheat Sheet

Your comprehensive guide to Probability Rules

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Descriptive Statistics

Sample Mean:
\[\bar x = \frac{\sum x_i}{n}\] Population Mean:
\[\mu = \frac{\sum x_i}{N}\]

Population Variance:
\[\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}\] Sample Variance:
\[s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}\]

Sample Standard Deviation:
\[s = \sqrt{\frac{\sum (x_i - \bar x)^2}{n-1}}\] Population Standard Deviation:
\[\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}\]

Percentile: Value below which a certain percentage of data falls
Quartiles: Q1 (25th percentile), Q2 = Median (50th percentile), Q3 (75th percentile)
Interquartile Range (IQR): \(Q_3 - Q_1\)
Range: Maximum - Minimum

Distribution Shapes: - Symmetric: Mean \(\approx\) Median - Right-skewed: Mean \(>\) Median (tail extends to the right) - Left-skewed: Mean \(<\) Median (tail extends to the left) - Outliers affect the mean more than the median

Probability Rules

Basic Probability Concepts

Probability Definition: \[P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\]

Properties: - \(0 \leq P(A) \leq 1\) - \(P(\emptyset) = 0\) (impossible event) - \(P(S) = 1\) (certain event, where \(S\) is sample space)

Example: Rolling a fair die, probability of getting an even number:
\[P(\text{even}) = \frac{3}{6} = \frac{1}{2} = 0.5\]

Practice: What is the probability of drawing a face card from a standard deck?
Answer: \(P(\text{face card}) = \frac{12}{52} = \frac{3}{13}\)

Complement Rule

Formula: \[P(A^c) = 1 - P(A)\] Alternative notation: \(P(A') = 1 - P(A)\)

Explanation: The probability that event \(A\) does not occur.

If the probability that Anya will graduate is 0.9, then the probability she will not graduate is:
\[P(\text{not graduate}) = 1 - 0.9 = 0.1\]

If \(P(\text{rain}) = 0.3\), what is \(P(\text{no rain})\)?
Answer: \(P(\text{no rain}) = 1 - 0.3 = 0.7\)

Addition Rules

General Addition Rule (For Any Two Events)

Formula: \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]

Explanation: We subtract \(P(A \cap B)\) to avoid double-counting the overlap.

In a class of 24 students, 10 are girls, 11 are A students, and 6 are girls who are A students.
Probability of selecting a girl or an A student:
\[P(\text{girl or A}) = \frac{10}{24} + \frac{11}{24} - \frac{6}{24} = \frac{15}{24} = 0.625\]

Addition Rule for Mutually Exclusive Events

Formula: \[P(A \cup B) = P(A) + P(B)\] Condition: \(P(A \cap B) = 0\) (events cannot occur simultaneously)

Probability of rolling a 2 or 6 on a die:
\[P(2 \text{ or } 6) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = 0.333\]

A bag contains 4 red, 3 blue, and 2 green marbles. What’s the probability of drawing a red or green marble?
Answer: \(P(\text{red or green}) = \frac{4}{9} + \frac{2}{9} = \frac{6}{9} = \frac{2}{3}\)

Multiplication Rules

Multiplication Rule for Dependent Events

Formula: \[P(A \cap B) = P(A) \times P(B|A)\] Alternative: \(P(A \cap B) = P(B) \times P(A|B)\)

Drawing two red cards without replacement from a standard deck:
\[P(\text{red and red}) = \frac{26}{52} \times \frac{25}{51} = 0.245\]

Multiplication Rule for Independent Events

Formula: \[P(A \cap B) = P(A) \times P(B)\] Condition: Events are independent if \(P(A|B) = P(A)\)

Drawing two red cards with replacement:
\[P(\text{red and red}) = \frac{26}{52} \times \frac{26}{52} = 0.25\]

Two fair coins are flipped. What’s the probability of getting two heads?
Answer: \(P(HH) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\)

Conditional Probability

Formula: \[P(A|B) = \frac{P(A \cap B)}{P(B)}\] Condition: \(P(B) > 0\)

Explanation: The probability of event \(A\) occurring given that event \(B\) has occurred.

In a group of 100 people, 60 are employed and 40 are unemployed. Of the employed, 45 are satisfied with their job.
What’s the probability someone is satisfied given they are employed?
\[P(\text{satisfied} | \text{employed}) = \frac{45}{60} = 0.75\]

A card is drawn from a deck. Given that it’s red, what’s the probability it’s a heart?
Answer: \(P(\text{heart} | \text{red}) = \frac{13}{26} = \frac{1}{2}\)

Set Operations and Probability

Union (OR): - Symbol: \(A \cup B\) - Meaning: Event \(A\) OR event \(B\) (or both) occurs - Formula: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

Intersection (AND): - Symbol: \(A \cap B\) - Meaning: Both events \(A\) AND \(B\) occur - Formula: \(P(A \cap B) = P(A) \times P(B|A)\)

Complement (NOT): - Symbol: \(A^c\) or \(A'\) - Meaning: Event \(A\) does NOT occur - Formula: \(P(A^c) = 1 - P(A)\)