Independent Events: Coins

P(A ∩ B) = P(A) × P(B)

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Two Heads in a RowTwo Heads Specific Sequence (HTHT)Sequence Multiple Coin FlipsMulti-Flip

Multiple Coin Flips

The general pattern for n flips:

Each flip has 2 possible outcomes. With n flips, you have 2 × 2 × ... × 2 = 2ⁿ total sequences. Since each flip is independent and fair (50/50), every sequence is equally likely.

Why is any specific sequence probability (½)ⁿ?

For each flip, you need a specific outcome (either H or T). Each has probability ½. Multiply n times:

This gives us the formula:

General Formula

Probability by Number of Flips

Flips	Total Outcomes	P(any specific sequence)
2	4
3	8
4	16
5	32
10	1,024
n	2^n

Example: 10 Heads in a Row

Probability:

Warning: Gambler's Fallacy

After flipping 9 heads in a row, many people think tails is "due". This is wrong! Each flip is independent. The 10th flip still has a 50/50 chance. The coin does not know what happened before.