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If a fair die is rolled 5 times, what is the probability, rounded to the nearest thousandth, of getting at most 1 four? Answer:

If a fair die is rolled 55 times, what is the probability, rounded to the nearest thousandth, of getting at most 11 four?\newlineAnswer:

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Q. If a fair die is rolled 55 times, what is the probability, rounded to the nearest thousandth, of getting at most 11 four?\newlineAnswer:
  1. Determine Probability of Rolling: Determine the probability of rolling a four on a single die roll.\newlineA fair die has 66 sides, so the probability of rolling any specific number, including a four, is 16\frac{1}{6}. Conversely, the probability of not rolling a four is 56\frac{5}{6}.
  2. Calculate Probability of One Four: Calculate the probability of rolling exactly one four in five rolls.\newlineThis can happen in several different ways: the four can appear on the first, second, third, fourth, or fifth roll. We will use the binomial probability formula, which is P(X=k)=(nk)⋅(pk)⋅((1−p)(n−k))P(X=k) = \binom{n}{k} \cdot (p^k) \cdot ((1-p)^{(n-k)}), where nn is the number of trials, kk is the number of successful outcomes we want, and pp is the probability of success on a single trial.\newlineFor exactly one four, k=1k=1, n=5n=5, and p=16p=\frac{1}{6}. So we calculate:\newlineP(X=1)=(51)⋅(16)1⋅(56)5−1P(X=1) = \binom{5}{1} \cdot (\frac{1}{6})^1 \cdot (\frac{5}{6})^{5-1}\newlineP(X=1)=5⋅(16)⋅(56)4P(X=1) = 5 \cdot (\frac{1}{6}) \cdot (\frac{5}{6})^4
  3. Calculate Probability of No Fours: Calculate the probability of rolling no fours in five rolls.\newlineFor this scenario, k=0k=0, n=5n=5, and p=16p=\frac{1}{6}. We use the binomial probability formula again:\newlineP(X=0)=(50)×(16)0×(56)5−0P(X=0) = \binom{5}{0} \times \left(\frac{1}{6}\right)^0 \times \left(\frac{5}{6}\right)^{5-0}\newlineP(X=0)=1×1×(56)5P(X=0) = 1 \times 1 \times \left(\frac{5}{6}\right)^5
  4. Add Probabilities for At Most One Four: Add the probabilities of rolling exactly one four and no fours to get the total probability of rolling at most one four.\newlineP(at most one 4)=P(X=0)+P(X=1)P(\text{at most one } 4) = P(X=0) + P(X=1)\newlineP(at most one 4)=(56)5+5⋅(16)⋅(56)4P(\text{at most one } 4) = (\frac{5}{6})^5 + 5 \cdot (\frac{1}{6}) \cdot (\frac{5}{6})^4
  5. Perform Calculations: Perform the calculations.\newlineP(X=0)=(56)5=(56)5≈0.40188P(X=0) = \left(\frac{5}{6}\right)^5 = \left(\frac{5}{6}\right)^5 \approx 0.40188\newlineP(X=1)=5×(16)×(56)4≈0.40188P(X=1) = 5 \times \left(\frac{1}{6}\right) \times \left(\frac{5}{6}\right)^4 \approx 0.40188\newlineP(at most one 4)=0.40188+0.40188≈0.80376P(\text{at most one 4}) = 0.40188 + 0.40188 \approx 0.80376
  6. Recalculate Probability of One Four: Step 55 (Correction): Perform the correct calculations.\newlineP(X=0)=(56)5≈0.40188P(X=0) = (\frac{5}{6})^5 \approx 0.40188 (corrected value)\newlineP(X=1)=5×(16)×(56)4≈0.40188P(X=1) = 5 \times (\frac{1}{6}) \times (\frac{5}{6})^4 \approx 0.40188 (incorrect value, needs to be recalculated)\newlineP(X=1)=5×(16)×(56)4=5×(16)×(6251296)≈0.32143P(X=1) = 5 \times (\frac{1}{6}) \times (\frac{5}{6})^4 = 5 \times (\frac{1}{6}) \times (\frac{625}{1296}) \approx 0.32143\newlineP(at most one 4)=P(X=0)+P(X=1)=0.40188+0.32143≈0.72331P(\text{at most one 4}) = P(X=0) + P(X=1) = 0.40188 + 0.32143 \approx 0.72331
  7. Round Final Probability: Round the final probability to the nearest thousandth. \newlineP(at most one 4)≈0.723P(\text{at most one } 4) \approx 0.723 (rounded to the nearest thousandth)

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