Read Binomial.pdfBinomial.pdf about the binomial distribution and the calculation of its probabilities in Excel. Also see Binomial.xlsx Download Binomial.xlsxfor examples of these probability calculations. Now suppose we roll a dice 60 times, and let X be the number of sixes that occur. Calculate P( X = 10 ) Calculate P( X < 10 ) Calculate P( X > 10 ) Calculate E( X )

The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial. These outcomes are labeled "success" and "failure". We are usually interested in the number of successes in a fixed number of trials.

In this case, rolling a six is considered a "success" and anything else is a "failure".

1. Calculate P( X = 10 )

The formula for the binomial probability is:

P(X = x) = nCx * (p^x) * ((1-p)^(n-x))

where:

• n is the number of trials
• x is the number of successes
• p is the probability of success on a single trial
• nCx is the number of combinations of n items taken x at a time

Here, n = 60 (number of trials), x = 10 (number of successes), and p = 1/6 (probability of rolling a six).

So, P(X = 10) = 60C10 * (1/6)^10 * (5/6)^50

2. Calculate P( X < 10 )

This is the sum of the probabilities of getting 0, 1, 2, ..., 9 sixes.

P(X < 10) = Σ P(X = x) for x from 0 to 9

3. Calculate P( X > 10 )

This is 1 minus the probability of getting 10 or fewer sixes.

P(X > 10) = 1 - P(X ≤ 10) = 1 - Σ P(X = x) for x from 0 to 10

4. Calculate E( X )

The expected value E(X) of a binomial distribution is n*p.

So, E(X) = 60 * 1/6 = 10

Note: The actual calculations of the probabilities can be done using Excel's BINOM.DIST function or a scientific calculator.