<s>[INST] Rubiks Cube Landing on Red A Rubik’s cube in which each side is painted one of six colors (white, orange, red, blue, green and yellow).
Suppose each side of the Rubik’s cube consists of only one color, if the Rubik’s cube is tossed 6 times what is the probability that the cube will land on the red side at least 4 times?
 [/INST] Add up the following:


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*The probability that it lands on the red side exactly $4$ times is $\dbinom{6}{4}\cdot\dfrac{5^2}{6^6}=\dfrac{375}{46656}$

*The probability that it lands on the red side exactly $5$ times is $\dbinom{6}{5}\cdot\dfrac{5^1}{6^6}=\dfrac{30}{46656}$

*The probability that it lands on the red side exactly $6$ times is $\dbinom{6}{6}\cdot\dfrac{5^0}{6^6}=\dfrac{1}{46656}$

Another way to look at it:


*

*The total number of ways it can land is $6^6=46656$

*The number of ways it can land on the red side exactly $4$ times is $\dbinom{6}{4}\cdot5^2=375$

*The number of ways it can land on the red side exactly $5$ times is $\dbinom{6}{5}\cdot5^1=30$

*The number of ways it can land on the red side exactly $6$ times is $\dbinom{6}{6}\cdot5^0=1$

In either case, the probability of it landing on the red side at least $4$ times is $\dfrac{375+30+1}{46656}$
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