Multiple choice

If a fair die is rolled $4$ times, then what is the probability that there are at least $2$ sixes ?

  1. $\dfrac{19}{144}$
  2. $\dfrac{25}{216}$
  3. $\dfrac{125}{216}$
  4. $\dfrac{175}{216}$
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A Correct answer
Explanation

Probability of at least 2 sixes = 1 - P(0 sixes) - P(1 six). P(0) = (5/6)^4 = 625/1296. P(1) = 4 * (1/6) * (5/6)^3 = 500/1296. Result = 1 - 1125/1296 = 171/1296 = 19/144.

AI explanation

Using the binomial distribution, the probability of getting exactly 0 sixes in 4 rolls is 4C0 multiplied by 1/6 to the power of 0 multiplied by 5/6 to the power of 4, which equals 625/1296. The probability of getting exactly 1 six is 4C1 multiplied by 1/6 multiplied by 5/6 to the power of 3, which equals 500/1296. The probability of getting at least 2 sixes is 1 minus the sum of these probabilities, calculated as 1 minus 1125/1296, which equals 171/1296. This simplifies to 19/144. The probability is 19/144.

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