GCSEC: Roll the Dice

ROLL THE DICE

There are 36 pair of dice in each box. Turn the box over two or three times to randomly roll the dice. Then hold the box at a slight angle and gently shake the box so that all of the die line up along one side of the box. How many times would you expect to find a two out of the 72 die? How about a six? If you group the die into 36 pair, how many times would you expect to find a sum of two out of the 36 pair? How about a sum of six?

If you would like to be part of this ongoing experiment, record your results on the accompanying charts.

When a single die is rolled, the outcome will be one of the six faces, and all are equally likely. Therefore on average when 72 die are rolled each of the six faces should turn up 12 times. As you will see on the accompanying chart for SINGLE DIE, the six faces are coming up about the same number of times, but not exactly. When the die are grouped into pairs, there are 36 possible outcomes: for example the first die could be a one and the second die a one, two, three, four, five or six; the first die could be a two and the second die a one, two, three, four, five or six; et cetera. Each of these outcomes are equally likely.
SUM OF A PAIR OF DICE

F i r s t

D i e

1 2 3 4 5 6

S
1 2 3 4 5 6 7

e
2 3 4 5 6 7 8

c D 3 4 5 6 7 8 9

o i 4 5 6 7 8 9 10

n e 5 6 7 8 9 10 11

d
6 7 8 9 10 11 12

However, there are eleven possible sums of the two die, and these sums are not equally likely. Out of the 36 equally likely outcomes, there is only one outcome that would produce a sum of two, namely where each of the two die shows the number one. Out of the 36 equally likely outcomes, there are five different ones that would produce a sum of six, namely one and five, two and four, three and three, four and two, and five and one. The possible sums of a pair of dice and the likelihood of each are:

SUM OF A PAIR OF DICE
			F	i	r	s	t
				D	i	e
			1	2	3	4	5	6
S		1	2	3	4	5	6	7
e		2	3	4	5	6	7	8
c	D	3	4	5	6	7	8	9
o	i	4	5	6	7	8	9	10
n	e	5	6	7	8	9	10	11
d		6	7	8	9	10	11	12

2	3	4	5	6	7	8	9	10	11	12
1/36	2/36	3/36	4/36	5/36	6/36	5/36	4/36	3/36	2/26	1/36

Random events such as rolling dice however usually do not match perfectly with their likelihoods or expected outcomes. A computer was used to "roll" a pair of dice 3,600,000 times. The expected number of times for the sum to be a two is 100,000 and the actual result was 100,092. The expected number of times for the sum to be a six is 500,000 and the actual result was 500,024.

The accompanying chart for PAIR OF DICE shows that this distribution is also close to what is expected, but not exactly the same.

Gambling operations, including state lotteries and casinos, are always designed with expected outcomes in mind. Let's say a game involved betting which number would come up when a die was rolled. The cost to play is $1 and the payoff for winning is $5.50. The bettor is willing to play the game 60 times, betting on the number 5 each time. Is the organization or the bettor likely to come out ahead?

Based upon the above explanation, you should know that on average the number five will come up once every six times, or ten times in 60. The cost to play 60 times is $60, and the expected payout is ten times $5.50 or $55. Therefore the bettor expects on average to lose about 8 cents each play. Of course, these are averages. In sixty rolls of a die the number 5 might come up ten times or thirty times, or no times.

You can make your own chart and track outcomes for flipping a coin or rolling a die.

If a coin is flipped ten times and comes up heads all ten times, what should be expected on the eleventh flip?

Previous flips have absolutely no effect on the outcome of the next flip. Regardless of the previous outcomes, whenever a coin is flipped, heads and tails are equally likely.

Coin flips, rolls of a die ... the actual results are never quite exactly the expected outcomes. What do the actual results look like over a large number of repetitions.

Flip a coin ten times and count the number of heads. You would expect five, but the number can be anything from zero to ten. Perhaps the actual result was four. Flip the coin another ten times and count the number of heads. This time the actual result might be five. Repeat this exercise one hundred times, each time recording the number of heads. Now create a graph of the possible outcomes (one through ten) against the actual results. You will see that the graph is highest for the outcome 5, and quickly tapers off in each direction.