A fair six-sided die is thrown \(N\) times and the result of each throw is recorded. (a) If the die is thrown \(N=12\) times, what is the probability that odd numbers occur three times? If it is thrown \(N=120\) times, what is the probability that odd numbers occur 30 times? Use the binomial distribution. (b) Compute the same quantities as in part (a) but use the Gaussian distribution. (Note: For part (a) compute your answers to four places.) (c) Plot the binomial and Gaussian distributions for \(N=2\) and \(N=12\).

The binomial distribution is used for probability calculations where there are a fixed number of independent trials, each with two possible outcomes: success or failure.
To calculate the probability that a specific number of successes occurs, we use the formula: \[ P(X = k) = \binom{N}{k} p^k (1-p)^{N-k} \] Here's a breakdown of the components:

• **N**: Number of trials
• **k**: Number of successes
• **p**: Probability of success on each trial
• **\(\binom{N}{k}\)**: Binomial coefficient, which counts the number of ways to choose k successes from N trials

To illustrate, consider a six-sided die rolled 12 times with the goal of finding the probability of getting an odd number 3 times:

• **N** = 12
• **k** = 3
• **p** = 0.5 (since half the sides of a die are odd numbers)

Using the binomial formula, we get: \[ P(X = 3) = \frac{12!}{3!(12-3)!} (0.5)^3 (0.5)^9 \approx 0.0537\] This method can be extended to compute probabilities for larger trials, e.g., 120 trials with 30 successes.

The Gaussian distribution, also known as the Normal distribution, is a continuous probability distribution that is symmetrical around its mean. It provides a good approximation for the binomial distribution when the number of trials (\(N\)) is large.
The Gaussian distribution formula is: \[ P(X = k) \approx \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(k - \mu)^2}{2\sigma^2}} \] Here:

• **\(\mu\)**: Mean of the distribution, given by \(Np\)
• **\(\sigma^2\)**: Variance of the distribution, given by \(Np(1-p)\)

For practical application, looking at the same die example:

• For **N = 12** and **k = 3**, with \( \mu = 12 \times 0.5 = 6\) and \( \sigma^2 = 12 \times 0.5 \times 0.5 = 3\), we use the Gaussian formula to get: \( P(X = 3) \approx 0.0807\)
• For **N = 120** and **k = 30**, with \( \mu = 60\) and \( \sigma^2 = 30\), the Gaussian formula gives us: \( P(X = 30) \approx 1.47 \times 10^{-7} \)

This shows that the Gaussian distribution can effectively approximate binomial probabilities for large trials.

Calculating probability is key in statistics and often involves defining the number of successful outcomes and total outcomes.
In the context of binomial and Gaussian distributions, probability calculations help in determining the likelihood of a certain number of successes (k) in a fixed number of trials (N).
Let's revisit the die example for a clear illustration:

• For **binomial distribution**, the calculation for getting 3 odd numbers out of 12 rolls when the probability of an odd number (success) is 0.5 is: \( P(X = 3) \approx 0.0537 \). This tells us there is about a 5.37% chance this event will occur.
• Using the **Gaussian approximation** for the same, we have: \( P(X = 3) \approx 0.0807 \) for 12 trials, showing a slightly different probability when approximated.

Probability calculations are similar for different distributions, but the methods vary. Hence, understanding the formulas and when to use each distribution is crucial.

Statistical approximation enables us to simplify complex probability problems. One common approximation in statistics is using the Gaussian (or Normal) approximation for the binomial distribution when the number of trials is large.
The reason behind using approximations is to make calculations more manageable, especially as values of N increase. Approximations are practical because:

• **Computational Simplicity**: Calculating factorials in the binomial coefficient (\( \binom{N}{k} \)) for large N can be cumbersome. Gaussian approximation is computationally less intensive.
• **Predictive Accuracy**: The Gaussian distribution often provides a close estimation of the binomial probabilities when N is large and p is not too close to 0 or 1.

Example comparison:

• For **N = 120** and **k = 30**, exact binomial probability: \( P(X = 30) \approx 0.0108 \)
• Gaussian approximation gives a very low probability: \( P(X = 30) \approx 1.47 \times 10^{-7} \)

While there's a big difference, the approximation still helps in understanding the general trend or likelihood of outcomes.

Plotting distributions is an excellent way to visualize and compare the probability of different outcomes under various statistical models.
Creating plots for binomial and Gaussian distributions helps to understand their similarities and differences better.
For example, let's consider the following cases:

• **N = 2**: - **Binomial Distribution**: Plot the probability for 0, 1, and 2 successes. - **Gaussian Distribution**: Not very relevant here as N is too small for an accurate approximation.
• **N = 12**: - **Binomial Distribution**: Plot the probability for getting different numbers of successes out of 12 trials. - **Gaussian Distribution**: Superimpose a normal curve using the same mean (\( \mu = Np \)) and variance (\( \sigma^2 = Np(1-p) \)).

By plotting, we can easily see that as N grows, the binomial distribution's shape approaches that of the Gaussian distribution. This visual representation helps in comprehending how statistical approximations function and where they are most effective.