A candy dish contains 30 jelly beans and 20 gumdrops. Ten candies are picked at random. What is the probability that 5 of the 10 are gumdrops? The two groups are jelly beans and gumdrops. Since the probability question asks for the probability of picking gumdrops, the group of interest (first group A in the formula) is gumdrops. The size of the group of interest (first group) is 30. The size of the second group is 20. The size of the sample is 10 (jelly beans or gumdrops). Let X = the number of gumdrops in the sample of 10. X takes on the values x = 0, 1, 2, ..., 10. What is the answer to the question "What is the probability of drawing 5 gumdrops in 10 picks from the dish?

A hypergeometric distribution is a discrete probability distribution that describes the probability of successes in draws with each draw being a success or failure, without replacement from a finite population size that contains exactly the same objects. The general formula is given as:

[tex]h(x) =\frac{({A}C_{x})({N-A}C_{n-x})}{^{N}C_{n} }[/tex]

where:

[tex]^{n}C_{r}=\frac{n!}{(n-r)!r!}[/tex]

h(x) is the probability of x successes,

n is the number of attempts,

A is the number of successes

N is the number of elements

For this problem:

A = 30, x = 5, N = 50, n = 10

The probability of drawing 5 gumdrops in 10 picks from the dish P(x=5) is

[tex]P(x=5)=\frac{({30}C_{5})({50-30}C_{10-5})}{^{50}C_{10} }\\P(x=5)=\frac{({30}C_{5})({20}C_{5})}{^{50}C_{10} }[/tex]

[tex]P(x=5)=\frac{\frac{30!}{(30-5)!5!}*\frac{20!}{(20-5)!5!} }{\frac{50!}{(50-10)!10!} }\\P(x=5)=\frac{\frac{30!}{25!5!}*\frac{20!}{15!5!} }{\frac{50!}{40!10!} }[/tex]

[tex]P(x=5)=\frac{142506*15504}{1.207*10^{10} }=0.215[/tex]

P(x=5) = 0.215

The probability of drawing 5 gumdrops in 10 picks from the dish is 0.215