Central Limit Theorem

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library(knitr)

Problem Statement

Problem 1 selected from page 352 of “Introduction to Probability”, by Charles M. Grinstead and J. Laurie Snell.

A die is rolled 24 times. Use the Central Limit Theorem to estimate the probability that (a) the sum is greater than 84. (b) the sum is equal to 84.

Answer:
(a)
Computation of Expected Value Ex1 and Variance Vx1 = (X - Ex1)^2, for one roll of die:

Ex1 <- (1 * (1/6) + 2 * (1/6) + 3 * (1/6) + 4 * (1/6) + 5 * (1/6) + 6 * (1/6))
Ex1

## [1] 3.5

Vx1 <- 1/6 * ((1 - Ex1)^2 + (2 - Ex1)^2 + (3 - Ex1)^2 + (4 - Ex1)^2 + (5 - Ex1)^2 + (6 - Ex1)^2)
Vx1

## [1] 2.916667

Computation of Expected Value Ex24, Variance Vx24 and Standard Deviation SD24, for 24 rolls of die:

Ex24 <- Ex1 * 24
Ex24

## [1] 84

Vx24 <- Vx1 * 24
Vx24

## [1] 70

SD24 <- sqrt(Vx24)
SD24

## [1] 8.3666

So, the Expected Value of 24 rolls is 84. The chances of being above or below this Expected Value are 50%-50% i.e. 83.5 or 84.5.

Now, we’ll compute the probability that the sum is greater than 84 i.e. P(sum > 84). With a correction of 0.5, it should be
P(sum > (84.5 - 84)/SD24) = P(sum > (84.5 - 84)/8.3666)

pnorm(84.5, 84, SD24, lower.tail = FALSE)

## [1] 0.4761728

Now, we’ll compute the probability that the sum equals 84 i.e. P((83.5 - 84)/8.3666) > sum > P((84.5 - 84)/8.3666)

pnorm(84.5, mean = 84, SD24) - pnorm(83.5, mean = 84, SD24)

## [1] 0.04765436

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