what is the probability of picking 2 silver or 2 gold?

Question

Lisa has 9 rings in her jewelery box. 5 are gold and 4 are silver. If she randomly selects 3 rings to wear to a party, find each probability.

Answer 1

Since you are only picking 3 rings, it is easy to go through all the cases to illustrate.

With the original 9 chances are 5/9 to pick a gold and 4/9 to pick a silver.

If you pick a gold first there are 4 gold out of 8 rings left so chances are 4/8 = 1/2 of picking the second gold. Since there are also 4 silver in the 8, chances are 1/2 of getting gold then silver.

If the first pick is a silver then the second pick gives you 5/8 for a gold or 3/8 for a silver since there are 5 out of 8 gold and 3 out of 8 silver left.

You can continue this for the third ring. If 2 gold are already picked it is 3/7 for another gold and 4/7 for a silver. If 2 silver are already picked it is 5/7 and 2/7. If a gold and silver have been picked it is 4/7 and 3/7

Multiply the probability of all eight possible selections:

ggg = 5/9 * 1/2 * 3/7 = 15/126
ggs = 5/9 * 1/2 * 4/7 = 20/126
gsg = 5/9 * 1/2 * 4/7 = 20/126
gss = 5/9 * 1/2 * 3/7 = 15/126
sgg = 4/9 * 5/8 * 4/7 = 20/126
sgs = 4/9 * 5/8 * 3/7 = 15/126
ssg = 4/9 * 3/8 * 5/7 = 15/126
sss = 4/9 * 3/8 * 2/7 = 6/126

As it should, the four cases do add up to 1.0 (always good to check). The chance of

Exactly 2 gold: ggs + gsg + sgg = 20/126 + 20/126 + 20/126 = 60/126

At least 2 gold = exactly 2 gold plus ggg = 60/126 + 15/126

Exactly 2 silver: gss + sgs + ssg = 15/126 + 15/126 + 15/126 = 45/126

At least 2 silver = exactly 2 silver + sss = 45/126 + 6/126 = 51/126

Answer 2

This is a hypergeometric distribution probability question. The probability of getting 2 silver is ((4 C 2)(5 C 1))/(9 C 3) = 0.3571; the probability of getting 2 gold is ((5 C 2)(4 C 1))/(9 C 3) = 0.4762, where C means to use the combination function, nCr on most scientific calculators.

Answer 3

The probability of drawing 2 gold is:

(5*4*4 + 5*4*4 + 4*5*4) / (9*8*7) = 10/21

The probability of drawing 2 silver is:

(4*3*5 + 4*5*3 + 5*4*3) / (9*8*7) = 5/14 = 7.5/21, showing that gold is more likely

Answer 4

hell... dump out the box, take them all and run

Answer 5

ask lisa.