Probability (for parents/teachers)

Roll of a single die:

A die is generally understood to be a cube with 6 faces labeled with the numbers 1, 2, 3, 4, 5, 6. Rolls of a die are considered to be independent and each face is considered to have equal probability for a fair die, namely:

P[X=1] = 1/6

P[X=2] = 1/6

P[X=3] = 1/6

P[X=4] = 1/6

P[X=5] = 1/6

P[X=6] = 1/6

For a single roll of a single die, only a single outcome is possible. For example, the probability that a single die thrown lands on a 3 and on a 6 in the same toss is 0 (these two events are said to be mutually exclusive because they can’t both happen at the same time). To understand rolls of two or more dice, we have to introduce the set of all possible outcomes, namely:

Die 1	Die 2
1	1
1	2
1	3
1	4
1	5
1	6
2	1
2	2
2	3
2	4
2	5
2	6
3	1
3	2
3	3
3	4
3	5
3	6
4	1
4	2
4	3
4	4
4	5
4	6
5	1
5	2
5	3
5	4
5	5
5	6
6	1
6	2
6	3
6	4
6	5
6	6

Here we understand that we have 36 possibilities (6 outcomes per die, crossed to make 36 possibilities). We can reasonable make statements about probability of specific events, for example:

Probability that the two faces are equal = 6 events / 36 possible events = 1/6

Probability that two faces are unequal = 30 events / 36 possible events = 5/6

These events can be counted based on our space of events defined above. Counting rules exist to make identification of event spaces and probabilities of events easier, but these rules typically require middle to high school math to apply. More advanced students may be able to understand the axioms of probability.

Considering the extension to 3 or more dice, we can reasonably use induction to enumerate our event space. We can use some straightforward calculations to find a couple of probabilities of interest. The total number of outcomes and probabilities for the lab events for various dice counts follows:

Dice	Outcomes	Count of half or more dice landing same face	P(half or more dice landing same face)	Count of all dice same face	P(all dice same face)
1	6
2	36	36	1	6	1/6
3	216			6	1/36
4	1296	216	1/6	6	1/216
5	7776			6	1/1296
6	46656	1296	1/36	6	1/7776

For six dice, to determine total possible outcomes: 6 possibilities for die 1 * 6 possibilities for die 2 * 6 possibilities for die 3 * 6 possibilities for die 4 * 6 possibilities for die 5 * 6 possibilities for die 6 = 46656 total events possible

For half or more dice to land on same face: 6 possibilities for die 1 * 1 possibility for die 2 * 1 possibility for die 3 * 6 possibilities for die 4 * 6 possibilities for die 5 * 6 possibilities for die 6 = 1296 events where this combination occurs. Consider that the independence of events allows us to interchange dice freely without loss of generality (we could say dice 1,2,3 need to land on same face the same way we could say dice 4, 3, and 6 need to land on same face. The assignment of the labels 1-6 is arbitrary).

Considering a single throw, with probability 1 out of 36 we would expect half or more of 6 dice to land on the same face. This is 216 times more likely than all dice landing on the same face. For repeated events, the probability of an event occurring 0 or more times would be governed by a binomial distribution with probability p. Considering a process of independent throws of 6 dice, here are expectations for the number of throws that meet the event criteria above for 6 dice (i.e. out of the number of throws, what is the probability of the event occurring in one or more throws?).

throws	p.half	p.all
1	2.78%	0.01%
2	5.48%	0.03%
3	8.10%	0.04%
4	10.66%	0.05%
5	13.14%	0.06%
6	15.55%	0.08%
7	17.90%	0.09%
8	20.18%	0.10%
9	22.39%	0.12%
10	24.55%	0.13%
25	50.55%	0.32%
50	75.55%	0.64%
75	87.91%	0.96%
100	94.02%	1.28%
200	99.64%	2.54%
500	100.00%	6.23%
1000	100.00%	12.07%
5000	100.00%	47.43%
10000	100.00%	72.37%
100000	100.00%	100.00%
1000000	100.00%	100.00%

Remember that the throws are independent, so a single realization of the process could have wildly different outcomes (it’s also worth noting that there is no force that would cause the long run expectation to correct to the probabilities presented above [i.e. with low probability these events might not actually happen given the number of throws above]. Another example of this phenomenon is the possibility of a significant differential in heads or tails appearing with repeated coin flips. Realizations of random processes are not guaranteed to match expectations. This difference is understood to be a fundamental failure for the “martingale” strategy for repeated wagers on independent events).

Students who carry out the experiment with 6 dice would find that they never (or almost never) see all dice land on the same face during a single throw of all dice. They might find a significantly larger number of occurrences (after repeated throws) of finding 3 or more dice land on the same face.

A useful mental exercise is to connect the probabilities of events such as dice throws, coin flips, and card draws to real world events. For example, how many dice would need to land on the same face to have an equivalent probability to being hit by lightning? How many dice to have an equivalent probability to making a basketball shot throwing backwards from half-court?

The next step is to consider other combinations of events and rules that govern those events (ex. what is the probability of 3 out of 6 dice landing on the same face, but having no single die land on a 4? This possibility happens to be 5 * 1 * 1 * 5 * 5 * 5 = 625 / 46656 ~ 1.3%). Students should also recognize that event spaces increase geometrically and not linearly (i.e. the change in outcomes for 3 dice instead of 2 is not the same as going to 6 from 5). Probability is used in almost every scientific field (particularly when connecting the fields of probability and statistics).

Probability problems occur frequently when considering wagers. Consider the following: “Mom wants me to load the dishwasher tonight, but I don’t really want to do it. She has offered to let me throw 6 dice and if they land on the same face, I don’t have to load the dishwasher. If two or more faces come up, I have to load the dishwasher for 2 weeks straight. Which outcome should I accept: a sure outcome of loading the dishwasher for one night, a 1/7776 chance of not having to load the dishwasher tonight, or a 7775/7776 chance (read “nearly sure”) chance of having to load the dishwasher for the next two weeks.” The rational choice is left to the reader/student to determine.

After considering event spaces with independent events (ex. independent dice throws), students can start to consider dependent events and conditional probability. For most introductory probability topics the mathematics are well settled, but many areas of inference are still undergoing heavy research (ex. search space for games such as Chess).