Question

Because of a mistake in​ packaging, a case of 16 bottles of wine contained 9 of brand A and 7 of brand​ B, each without labels. All the bottles look alike and have an equal probability of being chosen. Five bottles are randomly selected. ​(a) What is the probability that all five are brand​ A? ​(b) What is the probability that exactly two areare brand​ A? ​(c) What is the probability that none is brand​ A?

Answer 1

Answer:

a) The probability that all five are brand​ A is 0.0288

b) The probability that exactly two bottles are brand​ A is 0.0288

c) The probability that none of the bottles is brand​ A is 0.0048

Step-by-step explanation:

We have 9 bottles of brand A and 7 bottles of brand B.

The total of bottles is 16.

a) The probability that all five bottles are brand​ A is given by:

$P(5A)=\frac{9}{16} \frac{8}{15}\frac{7}{14} \frac{6}{13} \frac{5}{12}=\frac{3}{104}=0.0288$

b) Since we have 9 bottles of brand A we calculate the probability of picking two brand A bottles and the we calculate the probability of picking 3 brand B bottles:

$P(2A3B)=\frac{9}{16} \frac{8}{15}\frac{7}{14} \frac{6}{13} \frac{5}{12}=\frac{3}{104}=0.0288$

c) The probability that none of the bottles is brand​ A is the same as picking 5 brand B bottles:

$P(5B)=\frac{7}{16} \frac{6}{15}\frac{5}{14} \frac{4}{13} \frac{3}{12}=\frac{1}{208}=0.0048$