Question

Find the probabilities of getting the numbers from 1 to 6 upon rolling a die. Then find the probabilities of getting the numbers from 1 to 12 upon rolling two dice and summing the values that appear on their faces. Use technology to find and plot the corresponding probabilities for sums of 3, 4, 5, 10, 20, 50, and 100 dice. (a) As the number of dice increase, what shape does the probability distribution appear to approximate?

Answer 1

Single Dice:

If you throw a single dice, then it can fall six ways, each of which is equally likely if the dice is true. So the probability of getting one particular value is 1/6. If you want either of two values it is 2/6 or 1/3, and so on.

Two Dice:

It gets more interesting when you have two dice. One thing that you can do is work out what the total of the dice is. The dice experiment allows you to simulate throwing pairs of dice and see what the result is. This is a good introduction to probability, since you can see which combinations are more likely. But the real world, or even a simulated real world, never matches completely with calculated probability. So how do we calculate it? The first thing is to work out what the range is. You can't have a total less than 2 (both dice being 1) and you can't have a total more than 12 (both dice being 6). The easiest way to see what the probabilities is to write out the possible totals. There are 36 of them in all (6 x 6).

For more number of dice, to calculate the sum of the probabilities, I used R Studio. Package I used is dice.

Three Dice:

three_dice <- getSumProbs(ndicePerRoll = 3, nsidesPerDie = 6)$probabilities
plot(three_dice[, 1], three_dice[, 2], xlab = "Sum", ylab = "Probability", main = "Plot for three dice")

Four Dice:

four_dice <- getSumProbs(ndicePerRoll = 4, nsidesPerDie = 6)$probabilities
plot(four_dice[, 1], four_dice[, 2], xlab = "Sum", ylab = "Probability", main = "Plot for four dice")

Five Dice:

five_dice <- getSumProbs(ndicePerRoll = 5, nsidesPerDie = 6)$probabilities
plot(five_dice[, 1], five_dice[, 2], xlab = "Sum", ylab = "Probability", main = "Plot for five dice")

Ten Dice:

ten_dice <- getSumProbs(ndicePerRoll = 10, nsidesPerDie = 6)$probabilities
plot(ten_dice[, 1], ten_dice[, 2], xlab = "Sum", ylab = "Probability", main = "Plot for ten dice")
lines(ten_dice[, 1], ten_dice[, 2])

20 Dice:

twenty_dice <- getSumProbs(ndicePerRoll = 20, nsidesPerDie = 6)$probabilities
plot(twenty_dice[, 1], twenty_dice[, 2], xlab = "Sum", ylab = "Probability", main = "Plot for twenty dice")
lines(twenty_dice[, 1], twenty_dice[, 2])

50 Dice:

fifty_dice <- getSumProbs(ndicePerRoll = 50, nsidesPerDie = 6)$probabilities
plot(fifty_dice[, 1], fifty_dice[, 2], xlab = "Sum", ylab = "Probability", main = "Plot for fifty dice")
lines(fifty_dice[, 1], fifty_dice[, 2])

100 dice:

hundred_dice <- getSumProbs(ndicePerRoll = 100, nsidesPerDie = 6)$probabilities
plot(hundred_dice[, 1], hundred_dice[, 2], xlab = "Sum", ylab = "Probability", main = "Plot for hundred dice")
lines(hundred_dice[, 1], hundred_dice[, 2])

As the number increases, the distribution appears to approximate Normal (Gaussian) Distribution.