The shape
The normal distribution is the symmetric, bell-shaped curve that is highest in the middle and falls away smoothly on both sides. It is described by just two numbers: the mean, which is the centre, and the standard deviation, which sets how wide the bell is.
It shows up so often in probability for one main reason: many real quantities are sums or averages of lots of small independent contributions, and such sums tend to look like a bell.
Building a bell from dice
Start with one fair die. All six faces are equally likely, so the picture of possible results is flat. There is no bell at all.
Add a second die and sum them. Totals from 2 to 12 are possible, but they are not equally likely. Seven can happen six ways out of 36, while 2 and 12 can each happen only one way. The picture becomes a triangle, peaked in the middle. That is the reason behind the two dice totals pattern.
Keep adding dice. With three dice the peak is rounder and the tails thinner. By around ten dice the totals follow a bell curve so closely that the normal distribution is a good working description. This tendency of sums to approach a bell shape is the central limit theorem. It holds quite generally, provided the pieces are independent and none dominates the rest.
Numbers for ten dice
For one fair die the mean is 3.5 and the variance is 35/12, about 2.92. For ten independent dice:
- Mean of the total: 10 x 3.5 = 35.
- Variance: 10 x 35/12 = 350/12, about 29.17.
- Standard deviation: the square root of 29.17, about 5.4.
So the total of ten dice is centred on 35 with a spread of about 5.4. The shape would look much the same if you changed the number of faces; only the centre and width change.
The 68-95 rule of thumb
For results that follow a bell curve closely:
- About 68% fall within one standard deviation of the mean.
- About 95% fall within two standard deviations.
For ten dice, one standard deviation either side of 35 gives roughly 29.6 to 40.4. Two standard deviations gives roughly 24.2 to 45.8. So a total below 24 or above 46 would be uncommon, but not impossible. These are approximations; exact dice totals are whole numbers, and the real distribution is only close to the curve.
For coin flips, the number of heads in 100 flips has mean 50 and standard deviation 5. The rule suggests about 95% of results between roughly 40 and 60 heads. That is a useful sanity check: 56 heads is not strange, and 70 would be well outside what a fair coin usually gives.
The binomial link
The count of heads in repeated flips follows a binomial distribution. With many flips, its shape is almost indistinguishable from a normal curve, which is why the normal curve is used to approximate it. Counting methods such as Pascals triangle generate the exact heights for small cases, and you can see the bell emerge row by row.
Where the bell curve does not fit
A bell curve is not a universal law. It can mislead in several cases.
- Outcomes that are strongly skewed, such as a game with a small chance of a very large payout, do not look symmetrical, and the normal curve can understate the chance of extreme results.
- Results with a hard floor or cap, such as a stake that cannot fall below zero, deviate from the curve.
- Very small numbers of trials are too few to smooth out into a bell.
- When trials are not independent, the justification disappears.
It is also worth remembering that for skewed data the mean and the median can differ noticeably, so the centre of the bell is not always the “typical” result.
Why it matters when thinking about chance games
The bell curve gives a quick way to judge whether a session total is ordinary or unusual. For example, an even-money roulette bet repeated 100 times has a total with a standard deviation of about 10 units, whereas the expected loss from the house edge is only about 2.7 units. The curve is wide enough that finishing ahead is quite plausible. The curve also narrows relative to the expected loss as the number of bets rises, which is how the edge asserts itself over the long run.
A bell curve describes how results spread, not how to profit from them. Outcomes are uncertain, and a house edge works against the player on average. If gambling is causing difficulty, a local support service can help.