A single die is rolled 6 times. Find the probability of rolling a number less than three 3 times.
State answer as a decimal rounded to six decimal places.

Question

A single die is rolled 6 times. Find the probability of rolling a number less than three 3 times. State answer as a decimal rounded to six decimal places.

UpStudy AI · Accepted Answer

To find the probability of rolling a number less than three 3 times in 6 rolls of a single die, we can use the binomial probability formula.

The binomial probability formula is given by:
$$ P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k} $$

Where:
- $$ P(X = k) $$ is the probability of getting exactly $$ k $$ successes in $$ n $$ trials.
- $$ \binom{n}{k} $$ is the number of combinations of $$ n $$ items taken $$ k $$ at a time.
- $$ p $$ is the probability of success in a single trial.
- $$ n $$ is the total number of trials.

In this case, we have:
- $$ n = 6 $$ (6 rolls of the die)
- $$ k = 3 $$ (rolling a number less than three 3 times)
- $$ p = \frac{2}{3} $$ (probability of rolling a number less than three in a single roll)

Substituting these values into the binomial probability formula, we get:
$$ P(X = 3) = \binom{6}{3} \times \left(\frac{2}{3}\right)^3 \times \left(1-\frac{2}{3}\right)^{6-3} $$

Now, we can calculate the probability using the binomial probability formula.
Calculate the value by following steps:
- step0: Calculate:
  $$ { }_{6}C_{3}\left(\frac{2}{3}\right)^{3}\left(1-\frac{2}{3}\right)^{3}$$
- step1: Subtract the numbers:
  $$ { }_{6}C_{3}\left(\frac{2}{3}\right)^{3}\left(\frac{1}{3}\right)^{3}$$
- step2: Expand the expression:
  $$\frac{6!}{3!\times \left(6-3\right)!}\times \left(\frac{2}{3}\right)^{3}\left(\frac{1}{3}\right)^{3}$$
- step3: Subtract the numbers:
  $$\frac{6!}{3!\times 3!}\times \left(\frac{2}{3}\right)^{3}\left(\frac{1}{3}\right)^{3}$$
- step4: Reduce the fraction:
  $$\frac{6\times 5\times 4}{3!}\times \left(\frac{2}{3}\right)^{3}\left(\frac{1}{3}\right)^{3}$$
- step5: Multiply the terms:
  $$\frac{120}{3!}\times \left(\frac{2}{3}\right)^{3}\left(\frac{1}{3}\right)^{3}$$
- step6: Divide the terms:
  $$20\left(\frac{2}{3}\right)^{3}\left(\frac{1}{3}\right)^{3}$$
- step7: Multiply the terms:
  $$\frac{160}{27}\left(\frac{1}{3}\right)^{3}$$
- step8: Evaluate the power:
  $$\frac{160}{27}\times \frac{1}{3^{3}}$$
- step9: Multiply the fractions:
  $$\frac{160}{27\times 3^{3}}$$
- step10: Multiply:
  $$\frac{160}{3^{6}}$$
- step11: Evaluate the power:
  $$\frac{160}{729}$$
The probability of rolling a number less than three 3 times in 6 rolls of a single die is approximately 0.219479.
The probability of rolling a number less than three 3 times in 6 rolls of a single die is approximately 0.219479.

A single die is rolled 6 times. Find the probability of rolling a number less than three 3 times.
State answer as a decimal rounded to six decimal places.

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