a. P(the first flower is red) = "\\frac{10}{10 + 12} = \\frac{10}{22}"
P(the probability for the second flower to be red, if the first flower is red) = "\\frac{10-1}{9 + 12} = \\frac{9}{21}"
P(the third flower is red, if the first and the second flowers are red) = "\\frac{8}{20}"
P(the first 3 flowers are red) "= \\frac{10}{22} \\times \\frac{9}{21} \\times \\frac{8}{20} = \\frac{720}{9240} = 0.078" or 7.8 %
b. P(2 red and 2 white flowers) "= 6 \\times \\frac{10}{22} \\times \\frac{9}{21} \\times \\frac{12}{20} \\times \\frac{11}{19} = \\frac{71280}{175560} = 0.4060" or 40 %
c. At least 2 are red out of 4. The number of red roses can be 2, 3 or 4.
Total number of red roses are 10.
If 2 roses are red:
10C2 "\\times" 12C2 = "\\frac{10!}{8!2!} \\times \\frac{12!}{10!2!} = 2970"
If 3 roses are red:
10C3 "\\times" 12C1 = "\\frac{10!}{7!3!} \\times \\frac{12!}{11!} = 1440"
If 4 roses are red:
10C4 "\\times" 12C0 = "\\frac{10!}{6!4!} \\times 1 = 210"
P(at least 2 are red) "= \\frac{2970 + 1440 +210}{C^{22}_4} = \\frac{4620}{7315} = 0.631" or 63 %
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