By using the binomial probability formula, the probability of at least two 3's is equal to 0.1319.
Since the fair cubical die is thrown four times, the number of times is given by n = 4 and the probability of 3's is P = 1/6.
Mathematically, the binomial probability formula is given by:
[tex]P(X \geq x) =\sum^{n}_{r=x} ^nC_r (p)^r (q)^{(n-r)}\\\\P(X \geq 2) = \; ^4C_2 (\frac{1}{6} )^2 (\frac{5}{6} )^{(4-2)} + ^4C_3 (\frac{1}{6} )^3 (\frac{5}{6} )^{(4-3)} + ^4C_4 (\frac{1}{6} )^4 (\frac{5}{6} )^{(4-4)}\\\\P(X \geq 2) = 6 \times \frac{1}{36} \times \frac{25}{36} + 4 \times \frac{1}{216} \times \frac{5}{36} + 1 \times \frac{1}{1296} \times 1\\\\P(X \geq 2) = \frac{150}{1296} + \frac{20}{1296}+ \frac{1}{1296}[/tex]
P(X ≥ 2) = 19/144
P(X ≥ 2) = 0.1319.
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