5. You roll a dice 20 times,
a) what is the probability you roll a 6 exactly 8 of the 20
times?
b) what is the probability you roll a 5 or a 6 exactly 8 of
the 20 times? (Hint: define the probability of success on
each roll, then use the binomial distribution).


To solve these problems, we'll use the **binomial distribution**. A binomial distribution describes the number of successes in a fixed number of independent trials, each with the same probability of success. The probability of obtaining exactly \( k \) successes in \( n \) trials is given by:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

where:
- \( n = 20 \) is the number of trials (rolls).
- \( k = 8 \) is the number of successes (rolling a certain number exactly 8 times).
- \( p \) is the probability of success on each trial.
- \( \binom{n}{k} = \frac{n!}{k! (n-k)!} \) is the binomial coefficient.

### Part (a): Rolling a 6 Exactly 8 Times

**Step 1: Define the probability of success \( p \) for a single roll.**

The probability of rolling a 6 on a fair six-sided die is:

\[
p = \frac{1}{6}
\]

**Step 2: Apply the binomial distribution formula.**

Substitute the values into the formula:

\[
P(X = 8) = \binom{20}{8} \left( \frac{1}{6} \right)^8 \left( 1 - \frac{1}{6} \right)^{20 - 8}
\]

Simplify the expression:

\[
P(X = 8) = \binom{20}{8} \left( \frac{1}{6} \right)^8 \left( \frac{5}{6} \right)^{12}
\]

Let's compute this step-by-step.

### Part (b): Rolling a 5 or 6 Exactly 8 Times

**Step 1: Define the new probability of success \( p \) for each roll.**

Now, the success is rolling either a 5 or a 6. Since there are 2 favorable outcomes out of 6 possible outcomes:

\[
p = \frac{2}{6} = \frac{1}{3}
\]

**Step 2: Apply the binomial distribution formula.**

The probability of rolling a 5 or 6 exactly 8 times out of 20 is:

\[
P(Y = 8) = \binom{20}{8} \left( \frac{1}{3} \right)^8 \left( 1 - \frac{1}{3} \right)^{12}
\]

Simplify the expression:

\[
P(Y = 8) = \binom{20}{8} \left( \frac{1}{3} \right)^8 \left( \frac{2}{3} \right)^{12}
\]

Now, let's compute both probabilities.The probabilities are:

### Part (a)
The probability of rolling a 6 exactly 8 times in 20 rolls is approximately **0.0084** or **0.84%**.

### Part (b)
The probability of rolling a 5 or 6 exactly 8 times in 20 rolls is approximately **0.1480** or **14.80%**.

Would you like a more detailed explanation, or do you have any further questions?

### Related Questions:
1. What is the probability of rolling a 6 at least once in 20 rolls?
2. How would the probability change if the number of rolls was 30 instead of 20?
3. What is the expected number of times to roll a 6 in 20 rolls?
4. How do you calculate cumulative binomial probabilities?
5. How does the binomial distribution relate to the normal distribution?

**Tip:** The binomial distribution is useful in situations where you have a fixed number of independent trials and each trial has only two possible outcomes (success or failure).

You roll a dice 20 times, a) what is the probability you roll a 6 exactly 8 of the 20 times? b) what is the probability you roll a 5 or a 6 exactly 8 of the 20 times?

Explore the probability of rolling a 6 or a 5 exactly 8 times out of 20 dice rolls using the binomial distribution. Learn key probability concepts and formulas, with calculations of rolling a 6 or a 5 out of a fair die. Suitable for Grades 10-12 students.

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