Respuesta :
The probability of a selection of drinks is given by the number of required
ways divided by the number of possible outcome.
(a) The number of ways he can serve 3 bottles out of the 10 bottles of
zinfandel is 720 ways.
(b) The number of ways to select 6 bottles out of the 29 bottles is 475020 ways.
(c) The number of ways of selecting 2 bottles of each variety is 69,300 ways.
(d) The probability of selecting two bottles of each variety, when six bottles are selected is approximately 0.146
(e) The probability that six bottles are the same variety is approximately 0.001.
Reasons:
Number of bottles of zinfandel = 10
Number of bottles of merlot = 8
Number of bottles of cabernet = 11
(a) Number of bottles of zinfandel he wants to serve = 3 bottles
Number of ways of serving 3 bottles out of 10, with the order of serving being important is 10 permutation 3
[tex]^nP_r = \mathbf{\dfrac{n!}{(n - r)!}}[/tex]
Which gives;
[tex]^{10}P_3 = \mathbf{\dfrac{10!}{(10 - 3)!}} = 720[/tex]
- The number of ways he can serve 3 bottles out of the 10 bottles of zinfandel = 720 ways
(b) The number of ways to select 6 bottles out of the 29 bottles is given by combination, [tex]^{n}C_r[/tex] as follows;
[tex]^{29}C_6[/tex] = 475020
The number of ways to select 6 bottles out of the 29 bottles is 475020 ways.
(c) The number of ways 2 bottles can be selected from each variety is given as follows;
Number of ways = [tex]^{10}C_2[/tex] × [tex]^{8}C_2[/tex] × [tex]^{11}C_2[/tex] = 69,300
The number of ways of selecting 2 bottles of each variety = 69,300 ways
(d) The probability is given by the ratio of the number of ways of selecting 2 bottles from each variety to the number of ways of selecting 6 bottles from 29, as follows;
[tex]\displaystyle Probability = \frac{^{10}C_2 \times ^{8}C_2 \times ^{11}C_2}{^{29}C_6} = \frac{69,300}{475020} =\frac{55}{377} \approx 0.146[/tex]
The probability of selecting two bottles of each variety, when six bottles are selected P(2 each) ≈ 0.146
(e) The probability that six bottles are the same variety, is given by the sum
of the ways of selecting 6 bottles of a given variety, divided by the ways of
selecting six bottles from the 29 bottles as follows;
[tex]\displaystyle P(6 \ of \ same \ variety) = \frac{^{10}C_6 + ^{8}C_6 + ^{11}C_6}{^{29}C_6} = \frac{700}{475020} = \frac{5}{3393} \approx 1.474 \times 10^{-3}[/tex]
P(6 of same variety) ≈ 1.474 × 10⁻³ ≈ 0.001
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https://brainly.com/question/25297510
