Coin Toss

Consider the coin toss. Flip a coin; the result is heads (H) or tails (T).

Each time we flip it there are two possible outcomes with equal likelihood, so the probability of getting heads is 50% or 0.5 and the same is true for tails. But what if we want to know the probability of flipping twice and getting heads the first time and tails the second time? If we look at column B below, we can see all the possibilities. If we flip twice, there are four: heads-heads, heads-tails, tails-heads, and tails-tails. One of those equaly possible four outcomes is the one we're looking for, so the probibility is 1/4 = 0.25 or 25%. Note that for each flip the total number of possible outcomes (Column D) doubles. This is because for each of the previous outcomes we've added two variations, two branches on the probability tree we see in Column B. It turns out that if we want to know the probability of multiple events occurring, we can multiply the individual probabilities. So for if we want the chance of getting HT it's the chance of getting H first (1/2) times the chance of getting tails on the second flip (1/2), (1/2) * (1/2) = (1/4).

Okay, but what if we don't care about the order? What if we just want to know the chance of getting one heads and one tails in two flips? Either HT or TH counts, so it's 2/4 = 0.5. In Column C we can see the groupings of outcomes if we don't care about order. They present an interesting pattern. Look at the coefficients, the multiple of each item. They follow a pattern we call Pascal's Triangle.

Coefficients also follow Pascal's Triangle when we do binomial expansion. To find (A + B)³ we look at the fourth row of the triangle (we consider the first row to be more like the zeroth row) for the coefficients and get A³ + 3*(A²*B) + 3*(A*B²) + B³. But this is basically row three of our Column C above, but with A and B instead of H and T, if we say HHH is like H³. If we extend that metaphor, then flipping three times is like (H + T)³. It's interesting that the math seems to work on events like it does on numbers or variables. Really, that's the power of statistics and probability, doing math on events.