91影视

Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes鈥攈ow likely they are鈥攚hen we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

(a):

The probability mass function of multinomial distribution is

\(\begin{aligned}{{\rm{f}}_{{\rm{XYZ}}}}{\rm{(x,y,z) = P(X = x,Y = y,Z = z)}}\\{\rm{ = }}\frac{{{\rm{n!}}}}{{{\rm{x!y!z!}}}}{\rm{p}}_{\rm{1}}^{\rm{x}}{\rm{p}}_{\rm{2}}^{\rm{y}}{\rm{p}}_{\rm{3}}^{\rm{z}}{\rm{,}}\end{aligned}\)

where, analogously, this can be written in the same manner for given \({\rm{6}}\) random variables.

Given the probabilities

\(\begin{aligned}{*{20}{l}}{{{\rm{p}}_{\rm{1}}}{\rm{ = 0}}{\rm{.24,}}}&{{{\rm{p}}_{\rm{2}}}{\rm{ = 0}}{\rm{.13,}}}&{{{\rm{p}}_{\rm{3}}}{\rm{ = 0}}{\rm{.16,}}}\\{{{\rm{p}}_{\rm{4}}}{\rm{ = 0}}{\rm{.2,}}}&{{{\rm{p}}_{\rm{5}}}{\rm{ = 0}}{\rm{.13,}}}&{{{\rm{p}}_{\rm{6}}}{\rm{ = 0}}{\rm{.13,}}}\end{aligned}\)

where they sum to\({\rm{1}}\), the requested probability can be computed as follows

\({{\rm{f}}_{\rm{X}}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{,}}{{\rm{x}}_{\rm{3}}}{\rm{,}}{{\rm{x}}_{\rm{4}}}{\rm{,}}{{\rm{x}}_{\rm{5}}}{\rm{,}}{{\rm{x}}_{\rm{6}}}} \right){\rm{ = P}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ = }}{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{ = }}{{\rm{x}}_{\rm{2}}}{\rm{,}}{{\rm{X}}_{\rm{3}}}{\rm{ = }}{{\rm{x}}_{\rm{3}}}{\rm{,}}{{\rm{X}}_{\rm{4}}}{\rm{ = }}{{\rm{x}}_{\rm{4}}}{\rm{,}}{{\rm{X}}_{\rm{5}}}{\rm{ = }}{{\rm{x}}_{\rm{5}}}{\rm{,}}{{\rm{X}}_{\rm{6}}}{\rm{ = }}{{\rm{x}}_{\rm{6}}}} \right)\)

for \({{\rm{x}}_{\rm{1}}}{\rm{ = }}{{\rm{x}}_{\rm{2}}}{\rm{ = \ldots = }}{{\rm{x}}_{\rm{6}}}{\rm{ = 2}}\) is

\(\begin{aligned}{{\rm{f}}_{\rm{X}}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{,}}{{\rm{x}}_{\rm{3}}}{\rm{,}}{{\rm{x}}_{\rm{4}}}{\rm{,}}{{\rm{x}}_{\rm{5}}}{\rm{,}}{{\rm{x}}_{\rm{6}}}} \right)\\{\rm{ = }}\frac{{{\rm{12!}}}}{{{\rm{2!2!2!2!2!2!}}}}{\rm{ \times 0}}{\rm{.2}}{{\rm{4}}^{\rm{2}}}{\rm{ \times 0}}{\rm{.1}}{{\rm{3}}^{\rm{2}}}{\rm{ \times 0}}{\rm{.1}}{{\rm{6}}^{\rm{2}}}{\rm{ \times 0}}{\rm{.}}{{\rm{2}}^{\rm{2}}}{\rm{ \times 0}}{\rm{.1}}{{\rm{3}}^{\rm{2}}}{\rm{ \times 0}}{\rm{.1}}{{\rm{3}}^{\rm{2}}}{\rm{ = 0}}{\rm{.00213}}{\rm{.}}\end{aligned}\)

(b):

Binomial distribution with parameters \({\rm{n}}\) and \({\rm{p(q = 1 - p)}}\) has probability mass function

\({\rm{P(X = x) = }}\left( {\begin{array}{*{20}{l}}{\rm{n}}\\{\rm{x}}\end{array}} \right){{\rm{p}}^{\rm{x}}}{{\rm{q}}^{{\rm{n - x}}}}{\rm{,}}\;\;\;{\rm{xI \{ 0,1,2, \ldots n\} }}{\rm{.}}\)

The parameter \({\rm{p}}\) is the probability of success, therefore

\({{\rm{p}}_{\rm{4}}}{\rm{ = 0}}{\rm{.2}}\)

which means that

\({\rm{q = 1 - 0}}{\rm{.2 = 0}}{\rm{.8}}\)

The parameter \({\rm{n}}\) is 20. Hence, the requested probability is

\({\rm{P(X拢5) = }}\sum\limits_{{\rm{x = 0}}}^{\rm{5}} {\left( {\begin{aligned}{*{20}{c}}{{\rm{20}}}\\{\rm{x}}\end{aligned}} \right)} {\rm{0}}{\rm{.}}{{\rm{2}}^{\rm{x}}}{\rm{ \times 0}}{\rm{.}}{{\rm{8}}^{{\rm{20 - x}}}}{\rm{ = 0}}{\rm{.8042}}\)

(c):

Look at the blue, green, and orange candies as a success, and all other colors as failure. In this case the parameter \({\rm{p}}\) is

\({\rm{p = }}{{\rm{p}}_{\rm{1}}}{\rm{ + }}{{\rm{p}}_{\rm{3}}}{\rm{ + }}{{\rm{p}}_{\rm{4}}}{\rm{ = 0}}{\rm{.24 + 0}}{\rm{.16 + 0}}{\rm{.2 = 0}}{\rm{.6}}\)

which means that

\({\rm{q = 1 - 0}}{\rm{.6 = 0}}{\rm{.4}}\)

The parameter \({\rm{n}}\) is \({\rm{20}}\) . Hence, the requested probability is