A uniform distribution is flat. An experiment that can conform to a uniform distribution is any in which the probability of a single event is equal to that of all other events. One way to introduce students to this concept is to have them record die rolls. At first, students will see that the six categories of the histogram are uneven, but as time goes by, the shape will level out, becoming a single block.
It is easy to demonstrate the formation of the binomial distribution in class. A binomial distribution is a discrete distribution that describes the possibilities of two mutually exclusive events. One example of this is using coin tosses. Have students toss two coins and record the number of heads in the histogram. Students should find that after many trials, the category "1" is about twice the size of the category "0" or "2."
The normal distribution, also known as the Gaussian distribution or "bell curve," shows how a random trait or trial is likely to be closer to the average. One interesting way to show students the normal distribution is to repeat a binomial distribution, with some tweaks. Since the binomial distribution approaches the distribution of a normal distribution asymptotically, you can use a binomial distribution with a large number of trials. Thus, if needed, you can show students the formation of two distributions through one experiment. By drawing a smooth line over the binomial histogram, students will find that a normal distribution will appear.
The negative binomial distribution counts how many trials are needed for a number of successes. Coins are useful in showing the formation of this distribution. One example is to have students flip a coin until there are two "heads" results. At minimum, students will only need to flip a coin twice. There is no maximum number of flips. After recording the results in a histogram, students should find that the probabilities for seeing two heads are highest for 2 or 3 flips and get progressively lower. The shape of this histogram is the negative binomial distribution.