≡Problems involving probability
What are compound experiments?
They are experiments in which more than one action happens sequentially or simultaneously.
👉 Example:
Rolling a die twice
Tossing a coin twice
Drawing balls from an urn without replacement
1. Sample space in compound experiments
When the experiment is compound, the sample space increases.
📌 Example: tossing a coin twice
Possible outcomes:
(heads, heads)
(heads, tails)
(tails, heads)
(tails, tails)
👉 Total outcomes = 4
2. Probability Calculation
General Rule
\( P(A) = \frac{\text{favorable cases}}{\text{possible cases}} \)
📌 Example 1 — Coin
👉 Tossing a coin twice
Event: getting at least one head
Possible outcomes: #NL (heads, heads)
(heads, tails)
(tails, heads)
(tails, tails)
Favorable cases = 3
\( P(A) = \frac{3}{4} \)
📌 Example 2 — Dice
👉 Rolling a die twice
Event: sum equals 7
Cases Favorable:
(1,6)
(2,5)
(3,4)
(4,3)
(5,2)
(6,1)
Total possible cases:
\( 6 \times 6 = 36 \)
Favorable cases = 6
\( P(A) = \frac{6}{36} = \frac{1}{6} \)
3. Probability in experiments with dependence
👉 When the outcome of one event influences another.
📌 Example (without replacement)
Drawing two balls from a box without returning the first.
If there are 5 balls:
Probability changes after the first draw
4. Probability in Independent Experiments
👉 When one event does not affect the other.
📌 Example
Tossing a coin and rolling a die
\( P(A \cap B) = P(A) \times P(B)
📌 Example
Coin (heads) and die (even number):
\( P(\text{heads}) = \frac{1}{2}
\( P(\text{even}) = \frac{3}{6} = \frac{1}{2}
\( P(A \cap B) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
