91Ó°ÊÓ

The probability of success is a crucial concept in any probability problem. For a binomial distribution, success is the event we are interested in. In the context of rolling a die, success is rolling an ace, which means getting a 1. Since a standard die has six numbers (1 to 6), the ace's probability is the probability of rolling a 1 on a single trial. So, we calculate the probability of success as:

• The number of favorable outcomes (rolling a 1)
• Divided by the total number of possible outcomes (6 sides)

This gives us a probability of success, or rolling an ace, as \( \frac{1}{6} \). This probability will be used in calculating the chances of getting a specific number of successes in multiple trials.

Binomial distribution is vital when dealing with probability problems that involve two potential outcomes: success or failure. It applies to scenarios where we perform a number of independent trials, each with the same probability of success.

In our problem of rolling a die, the binomial distribution can be used since each roll is an independent trial. The odds of rolling an ace do not change based on previous rolls. The parameters of a binomial distribution here include:

• n: The total number of trials (6 rolls)
• p: The probability of success in each trial (chance of rolling an ace, \( \frac{1}{6} \))

Understanding binomial distribution helps us calculate the probability of achieving a specific number of successes within numerous attempts.

The binomial coefficient is a factor that helps calculate the number of ways to achieve a certain number of successes in a series of trials. It provides the combination count, denoted by \( \binom{n}{k} \), and is pivotal in the binomial probability formula.

To find \( \binom{6}{1} \), which is how many ways we can get exactly one ace in six rolls, we use:\[\binom{6}{1} = \frac{6!}{1!(6-1)!}\]Where '!' denotes factorial, meaning to multiply the number by all positive integers less than it. Thus, \( 6! \) is \( 6 \times 5 \times 4 \times 3 \times 2 \times 1 \). Simplifying this gives us:\[\binom{6}{1} = 6\]The binomial coefficient shows there are 6 ways to roll one ace in 6 die rolls.

Once we've determined all the components, we can calculate the overall probability using the binomial probability formula. Here's how we bring it all together:The formula is:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]Where:

• n is the number of trials (6 dice rolls)
• k is the number of successful outcomes we want (1 ace)
• p is the probability of success (\( \frac{1}{6} \))
• \( (1-p) \) is the probability of failure (\( \frac{5}{6} \))

Substitute these values into the formula:\[P(X = 1) = \binom{6}{1} \left( \frac{1}{6} \right)^1 \left( \frac{5}{6} \right)^{5}\]This simplifies to:\[P(X = 1) = 6 \times \frac{1}{6} \times \left( \frac{5}{6} \right)^5\]Further simplifying, you get \( P(X = 1) \approx 0.4019 \), which means around a 40.19% chance to roll exactly one ace in 6 rolls.