Answer:
Answer explained below
Step-by-step explanation:
(a)
Simulate the rolling of two dice 10,000 times D1 and D2 are the 10,000 results of roll of dice 1 and dice 2.
D1 = sample(c(1:6), 10000, replace = TRUE)
D2 = sample(c(1:6), 10000, replace = TRUE)
Sum = D1 + D2
(b)
The event A, the dice add up to a perfect square (4 or 9).
A = Sum[Sum == 4 | Sum ==9]
Proportion of A, P(A) = 0.189
length(A) / length(Sum)
(c)
The event B, the dice add up to an even number.
B = Sum[Sum %% 2 == 0]
Proportion of B, P(B) = 0.5049
length(B) / length(Sum)
(d)
The rolls are in A ∩ B (common to both A and B)
> intersect(A,B)
[1] 4
The proportion that are in A ∩ B is 0.0792
length(Sum[Sum == 4]) / length(Sum)
P(A) * P(B) = 0.189 * 0.5049 = 0.0954
The proportion in A multiplied by the proportion that are in B is not equal to P(A ∩ B)
(e)
Of the rolls in which B occurs, the proportion of those rolls are also in A is 0.4190476
length(Sum[Sum == 4]) / length(A)
This proportion is greater than the P(A) calculated in part (b).
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