Pascal's Triangle for Coin Toss Probability: Skip Listing All Outcomes

Tossing 5 coins and need the probability of exactly 2 heads? Or "at least 3 heads" when 6 coins are tossed? You don't need to write out all 32 or 64 outcomes. Pascal's Triangle gives you the answer in seconds. In this clip, you'll learn how to use Pascal's Triangle to find the number of ways of getting a given number of heads (or tails) when coins are tossed. This is the numerator of every coin toss probability question. What you'll learn in this clip: How the number of heads and the number of ways of getting them follow a pattern for 1, 2, 3, 4 and 5 coins Why the totals 2, 4, 8, 16 and 32 confirm that every possible outcome is covered How to build Pascal's Triangle step by step: 1 1 → 1 2 1 → 1 3 3 1 → 1 4 6 4 1 → 1 5 10 10 5 1 → 1 6 15 20 15 6 1 Why the rows look like powers of 11 (11², 11³, 11⁴) but the trick breaks down from n = 5, and why you must write the row as 1 5 10 10 5 1, not 1 6 1 0 5 1 How to find the denominator instantly using 2ⁿ, so you only have to work out the numerator Solved examples: exactly 2 heads with 4 coins (6/16 = 3/8), exactly 2 heads with 5 coins (10/32 = 5/16), 4 heads with 5 coins (5/32), and at least 3 heads with 6 coins (42/64 = 21/32) The complement shortcut: P(either/or) + P(neither/nor) = 1, so P(at least 3 heads) = 1 − P(0, 1 or 2 heads) How to handle "number of tails" questions by converting them to heads, or by reading the same table directly, since an unbiased coin treats heads and tails alike Why Pascal's Triangle is better than listing outcomes, but still slow when you need 8 coins and must build every row from the start Who is this for? Students learning probability for school exams, JEE, aptitude tests and entrance exams, where time is limited and listing 256 outcomes for 8 coins is not an option. Key takeaway: Pascal's Triangle gives you the number of ways to get 0, 1, 2, 3... heads. Divide by 2ⁿ for the probability. Use the complement trick for "at least" questions. This method is a big step up from listing outcomes, but it has limits. Watch to the end to see why a faster method is coming next. Watch the full clip, try the practice cases yourself (for example, exactly 4 tails with 5 coins), and save this video for revision before your next probability test. #PascalsTriangle #CoinTossProbability #Probability #NumberOfHeads #AtLeastThreeHeads #ProbabilityShortcut #ProbabilityTricks #MathsTricks #EntranceExamMaths