Question

10-A biased die is thrown thirty times and the number of sixes seen is eight. ( The probability of face six is 8 = 4 ) If the die is thrown a further twelve times find:
30 15
(a) the probability that a six will occur exactly twice; (b) the expected number of sixes: () = ;
(c) the variance of the number of sixes.

Answer 1

Using the binomial distribution, we have that:

a) There is a 0.2111 = 21.11% probability that a six will occur exactly twice.

b) The expected number of sixes is of 3.2.

c) The variance of the number of sixes is of 2.35.

What is the binomial distribution formula?

The formula is:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$C_{n,x} = \frac{n!}{x!(n-x)!}$

The parameters are:

• x is the number of successes.

• n is the number of trials.

• p is the probability of a success on a single trial.

For this problem, the parameters are given by:

p = 8/30 = 0.2667, n = 12.

Item a:

The probability is P(X = 2), hence;

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 2) = C_{12,2}.(0.2667)^{2}.(1-0.2667)^{10} = 0.2111$

Item b:

The expected number of the binomial distribution is:

E(X) = np.

Hence:

E(X) = 12 x 8/30 = 3.2.

Item c:

The variance of the binomial distribution is:

V(X) = np(1-p).

Hence:

E(X) = 12 x 8/30 x 22/30 = 2.35.

More can be learned about the binomial distribution at https://brainly.com/question/24863377

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