To calculate the likelihood of rolling four fives in six rolls of a die, we'll use the binomial probability formula:

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

Where:

• $n$ is the total number of trials (6 rolls),
• $k$ is the number of successes (4 fives),
• $p$ is the probability of rolling a five ($\frac{1}{6}$),
• $(1-p)$ is the probability of rolling anything else ($\frac{5}{6}$).

Step 1: Plug in the values.

$P(X = 4) = \binom{6}{4} \left(\frac{1}{6}\right)^4 \left(\frac{5}{6}\right)^2$

Step 2: Calculate the binomial coefficient $\binom{6}{4}$:

$\binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6 \cdot 5}{2 \cdot 1} = 15$

Step 3: Calculate the powers:

• $\left(\frac{1}{6}\right)^4 = \frac{1}{1296}$,
• $\left(\frac{5}{6}\right)^2 = \frac{25}{36}$.

Step 4: Combine everything:

$P(X = 4) = 15 \cdot \frac{1}{1296} \cdot \frac{25}{36}$

Simplify: $P(X = 4) = 15 \cdot \frac{25}{46656} = \frac{375}{46656} \approx 0.00804$

Step 5: Express as odds.

Odds are expressed as $\text{successes} : \text{failures}$. In this case: $\text{Odds} = 1 : \frac{1}{0.00804} \approx 1 : 124.4$

Thus, the odds are approximately 1 in 124.

Would you like further explanation or detailed steps?

Questions to expand on:

1. What is the binomial probability formula, and why is it used here?
2. How does the probability of rolling different numbers on a die affect the outcome?
3. What are some real-life applications of the binomial probability distribution?
4. How would the calculation change if the number of trials increased?
5. Can this result be confirmed using a simulation?

Tip:

When calculating probabilities for independent events, make sure to double-check that the total probabilities add up to 1. This ensures accuracy.