🎲 Probability – Advanced Topics
Probability tables, conditional probability, dependent/independent events, and the binomial distribution
📊 Part 1: Probability Table
A probability table displays all probabilities of two events in an organised format.
Table structure:
| \(B\) | \(\bar{B}\) | Total | |
|---|---|---|---|
| \(A\) | \(P(A \cap B)\) | \(P(A \cap \bar{B})\) | \(P(A)\) |
| \(\bar{A}\) | \(P(\bar{A} \cap B)\) | \(P(\bar{A} \cap \bar{B})\) | \(P(\bar{A})\) |
| Total | \(P(B)\) | \(P(\bar{B})\) | 1 |
💡 Rules:
- Sum of each row = row probability (in the Total column)
- Sum of each column = column probability (in the Total row)
- Sum of the entire table = 1
✏️ Example: Survey on Sport and TV Watching
A survey found: 40% do sport, 60% watch TV, 25% do both.
| TV (T) | No TV | Total | |
|---|---|---|---|
| Sport (S) | 0.25 | 0.15 | 0.40 |
| No Sport | 0.35 | 0.25 | 0.60 |
| Total | 0.60 | 0.40 | 1 |
How we filled it in:
\(P(S \cap \bar{T}) = P(S) - P(S \cap T) = 0.40 - 0.25 = 0.15\)
\(P(\bar{S} \cap T) = P(T) - P(S \cap T) = 0.60 - 0.25 = 0.35\)
\(P(\bar{S} \cap \bar{T}) = 1 - 0.25 - 0.15 - 0.35 = 0.25\)
🎯 Part 2: Conditional Probability
Conditional probability is the probability that an event will occur, given that another event has already occurred.
Core formula:
\(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
The probability of \(A\) given that \(B\) has occurred
💡 How to read it: \(P(A|B)\) = "the probability of A given B"
The vertical bar | is read as "given" or "given that"
✏️ Example: Using the Previous Table
Question: What is the probability that a person does sport, given that they watch TV?
\(P(S|T) = \frac{P(S \cap T)}{P(T)} = \frac{0.25}{0.60} = \frac{25}{60} = \frac{5}{12} \approx 0.417\)
Question: What is the probability that a person watches TV, given that they do sport?
\(P(T|S) = \frac{P(S \cap T)}{P(S)} = \frac{0.25}{0.40} = \frac{25}{40} = \frac{5}{8} = 0.625\)
⚠️ Note: \(P(A|B) \neq P(B|A)\) in general!
🔄 Multiplication Rule (from the formula):
\(P(A \cap B) = P(B) \cdot P(A|B) = P(A) \cdot P(B|A)\)
💡 Useful: This is exactly what we do in a probability tree!
When multiplying along a path, we are actually multiplying conditional probabilities.
🔗 Part 3: Dependent and Independent Events
✅ Independent Events
The occurrence of one does not affect the other
Condition:
\(P(A|B) = P(A)\)
or equivalently:
\(P(A \cap B) = P(A) \cdot P(B)\)
Examples:
- Rolling two dice
- Tossing a coin twice
- Drawing with replacement
❌ Dependent Events
The occurrence of one affects the other
Condition:
\(P(A|B) \neq P(A)\)
or equivalently:
\(P(A \cap B) \neq P(A) \cdot P(B)\)
Examples:
- Drawing without replacement
- Selecting cards from a deck
- Choosing people from a group
⊘ Mutually Exclusive Events
Cannot occur at the same time!
\(P(A \cap B) = 0\)
Example:
When rolling a die: "got 1" and "got 6" are mutually exclusive
Formula:
\(P(A \cup B) = P(A) + P(B)\)
(no overlap)
⚠️ Don't confuse!
Mutually exclusive ≠ independent!
Mutually exclusive events are actually dependent (if one occurs, the other certainly cannot)
📊 Comparison Table:
| Independent | Mutually exclusive | |
|---|---|---|
| Meaning | Do not affect each other | Cannot occur simultaneously |
| \(P(A \cap B)\) | \(P(A) \cdot P(B)\) | \(0\) |
| \(P(A \cup B)\) | \(P(A) + P(B) - P(A)P(B)\) | \(P(A) + P(B)\) |
📈 Part 4: Binomial Distribution
The binomial distribution describes an experiment with only two outcomes (success/failure) that is repeated a number of times.
Conditions for a binomial distribution:
1️⃣
\(n\) trials
Fixed number of repetitions
2️⃣
Two outcomes
Success or failure
3️⃣
Constant \(p\)
Same success probability
4️⃣
Independent
Trials are independent
⭐ The Binomial Formula:
\(P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}\)
| \(n\) | Number of trials |
| \(k\) | Desired number of successes |
| \(p\) | Probability of success in a single trial |
| \(1-p\) | Probability of failure in a single trial (sometimes denoted \(q\)) |
| \(\binom{n}{k}\) | Binomial coefficient – number of ways to choose \(k\) from \(n\) |
📐 Reminder – Binomial coefficient:
\(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)
✏️ Detailed Example:
Question: A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?
Identify the values:
- \(n = 5\) (5 tosses)
- \(k = 3\) (want 3 heads)
- \(p = 0.5\) (probability of heads)
Substitute into the formula:
\(P(X = 3) = \binom{5}{3} \cdot (0.5)^3 \cdot (0.5)^{5-3}\)
\(= \binom{5}{3} \cdot (0.5)^3 \cdot (0.5)^2\)
\(= 10 \cdot 0.125 \cdot 0.25\)
\(= 10 \cdot 0.03125\)
\(= 0.3125 = \frac{5}{16}\)
💡 Explanation:
\(\binom{5}{3} = 10\) = there are 10 ways to choose which 3 out of 5 tosses give heads
\((0.5)^3\) = probability of 3 heads
\((0.5)^2\) = probability of 2 tails
🧮 Expected Value and Variance:
Expected value (mean)
\(E(X) = n \cdot p\)
Variance
\(Var(X) = n \cdot p \cdot (1-p)\)
Standard deviation
\(\sigma = \sqrt{np(1-p)}\)
Example: In 100 coin tosses (\(n=100, p=0.5\)):
Expected value: \(E(X) = 100 \cdot 0.5 = 50\) (expect 50 heads)
Standard deviation: \(\sigma = \sqrt{100 \cdot 0.5 \cdot 0.5} = \sqrt{25} = 5\)
📝 Common Binomial Question Types:
| Question type | How to calculate |
|---|---|
| Exactly \(k\) | \(P(X = k)\) |
| At most \(k\) | \(P(X \leq k) = \sum_{i=0}^{k} P(X=i)\) |
| At least \(k\) | \(P(X \geq k) = 1 - P(X \leq k-1)\) |
| At least one | \(P(X \geq 1) = 1 - P(X = 0)\) |
| None | \(P(X = 0) = (1-p)^n\) |
📝 Summary of Formulas
| Conditional probability: | \(P(A|B) = \frac{P(A \cap B)}{P(B)}\) |
| Independent: | \(P(A \cap B) = P(A) \cdot P(B)\) |
| Mutually exclusive: | \(P(A \cup B) = P(A) + P(B)\) |
| Binomial: | \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\) |