What is the probability of rolling exactly one 3 when a fair 6-sided die is rolled three times?

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To solve the problem of finding the probability of rolling exactly one ‘3’ when a fair 6-sided die is rolled three times, we can apply the binomial probability formula. The situation can be modeled as a binomial experiment, where:

  • ‘n’ (the number of trials) = 3 rolls of the die
  • ‘k’ (the number of successes we are interested in) = 1 (rolling one ‘3’)
  • ‘p’ (the probability of success on each trial) = 1/6 (since there is one ‘3’ and six faces on a die)
  • ‘q’ (the probability of failure on each trial) = 5/6 (the probability of rolling any number other than ‘3’)

The binomial probability formula is given by:

P(X = k) = C(n, k) * (pk) * (q(n-k))

Where:

  • C(n, k) is the binomial coefficient, calculated as C(n, k) = n! / (k! * (n – k)!)

Now, substituting the values into the formula:

  1. Calculate the binomial coefficient:

    2. C(3, 1) = 3! / (1! * (3 – 1)!) = 3 / 1 = 3

  2. Plug the values into the probability formula:

    3. P(X = 1) = C(3, 1) * (1/6)1 * (5/6)(3-1)

    P(X = 1) = 3 * (1/6) * (5/6)2

    P(X = 1) = 3 * (1/6) * (25/36)

    P(X = 1) = 3 * (25/216) = 75/216

    P(X = 1) = 25/72

So, the probability of rolling exactly one ‘3’ when rolling a fair 6-sided die three times is 25/72 or approximately 0.3472.

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