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 )

This is a binomial distribution problem. In a binomial distribution, the probability of getting exactly k successes (defined as the outcome we are interested in, in this case rolling a six) in n trials (in this case 60 dice rolls) is given by the formula:

P(X = k) = C(n, k) * (p^k) * ((1-p)^(n-k))

where:

• C(n, k) is "n choose k", the number of combinations of n items taken k at a time
• p is the probability of success on each trial (in this case 1/6, the probability of rolling a six)
• n is the number of trials (in this case 60)
• k is the number of successes we are interested in

Step 1: Calculate P(X = 10)

P(X = 10) = C(60, 10) * ((1/6)^10) * ((5/6)^(60-10)) = 75394027566 * (1.65381716859e-8) * (0.000002032) = 0.025

Step 2: Calculate P(X < 10)

P(X < 10) = P(X = 0) + P(X = 1) + ... + P(X = 9)

We would calculate each of these probabilities using the same formula as in Step 1, but with different values of k. However, this would be quite tedious. Instead, we can use a binomial cumulative distribution function (CDF) to calculate this directly. Most statistical software or calculators can do this.

P(X < 10) = binomial_CDF(n=60, p=1/6, k=9) = 0.18 (approx)

Step 3: Calculate P(X > 10)

P(X > 10) = 1 - P(X <= 10) = 1 - [P(X = 0) + P(X = 1) + ... + P(X = 10)] = 1 - binomial_CDF(n=60, p=1/6, k=10) = 1 - 0.30 (approx) = 0.70 (approx)

Step 4: Calculate E(X)

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

E(X) = n*p = 60 * (1/6) = 10

So, P(X = 10) = 0.025, P(X < 10) = 0.18, P(X > 10) = 0.70, and E(X) = 10.