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).
