5 Februa 2016 You have twenty bottles of wine in your wine c

5. Februa 2016 You have twenty bottles of wine in your wine cellar, Ten are Cabernet, six are Merlot, and four are syrah. You randomly select four bottles. Find the probability they are all of the same varietal That is, they are all Cabernet or all Merlot or all Syrah.

Solution

Solution:

Its a problem of combinations in probability.

A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected.

For example, suppose we have a set of three letters: A, B, and C. We might ask how many ways we can select 2 letters from that set. Each possible selection would be an example of a combination. The complete list of possible selections would be: AB, AC, and B

there are 20 bottles of wine=10 cabernet+6 merlot+4 syrah.

randomly selecting 4 bottles out of 20 is

20_C4=4845 ways.

I can

pick 4 cabernet out of 10 cabernet is 10_C 4=210 ways

pick 4 merlot out of 6 is 6_C4=15 ways

pick 4 syrah out of 4 syrah is 4_C4= 1 ways

There are 210+15+1=226 ways to pick 4 bottles of same variety.

probability =favourable ways/total ways=226/4845=0.0466

=4.66%