Answer:
a
[tex]P(N_g) = 0.5155[/tex]
b
[tex]P(K ) = 0.36364[/tex]
c
[tex]P(U1 | N_y ) = 0.7223[/tex]
Step-by-step explanation:
From the question we are told that
The number of green marbles in Urn 1 is [tex]N_g = 4[/tex]
The number of yellow marbles in Urn 1 is [tex]N_y = 5[/tex]
The number of green marbles in Urn 2 is [tex]n_g = 7[/tex]
The number of yellow marbles in Urn 1 is [tex]n_y = 4[/tex]
The probability of choosing Urn 1 is [tex]P(U1) = 0.63[/tex]
The probability of choosing Urn 2 is [tex]P(U2) = 1- 0.63 =0.37[/tex]
Generally the total marble in Urn 1 is
[tex]N_t = N_g +N_y[/tex]
=> [tex]N_t = 4 + 5[/tex]
=> [tex]N_t = 9[/tex]
Generally the total marble in Urn 2 is
[tex]n_t = n_g +n_y[/tex]
=> [tex]n_t = 7+4[/tex]
=> [tex]n_t = 11[/tex]
Generally the probability of choosing green marble is
[tex]P(N_g) = P(U1 ) * \frac{N_g}{N_t} + [P(U2 ) * \frac{n_g}{n_t} ][/tex]
=> [tex]P(N_g) = 0.63 * \frac{4}{9} + [0.37 * \frac{7}{11} ][/tex]
=> [tex]P(N_g) = 0.5155[/tex]
Generally the probability that a yellow marble was chosen, if it is known that Urn 2 was chosen is mathematically represented as
[tex]P(K ) = \frac{n_y}{n_t}[/tex]
=> [tex]P(K ) = \frac{4}{11}[/tex]
=> [tex]P(K ) = 0.36364[/tex]
Generally the probability of choosing yellow marble is
[tex]P(N_y) = P(U1 ) * \frac{N_y}{N_t} + [P(U2 ) * \frac{n_y}{n_t} ][/tex]
=> [tex]P(N_y) = 0.63 * \frac{5}{9} + [0.37 * \frac{4}{11} ][/tex]
=> [tex]P(N_y) = 0.4845[/tex]
Generally the probability that Urn 1 was chosen, if it is known that a yellow marble was drawn is mathematically represented as
[tex]P(U1 | N_y ) = \frac{ P( U1 \ n N_y)}{P(N_y)}[/tex]
=> [tex]P(U1 | N_y ) = \frac{0.63 * [\frac{5}{9} ] }{0.4845 }[/tex]
=> [tex]P(U1 | N_y ) = 0.7223[/tex]
