## Sunday, July 08, 2007

### My Dreams are Riddles, Riddled with Math

A question on YA: "There are 120 customers in a store, 80 purchased at least 1 item. If 4 people are randomly selected, what's the probability all will buy at least 1 item?"

My Answer: The probability of any one customer having purchased an item is 80 out of 120 or 2/3. So if we are looking at parts of a whole, that represent all the possible combinations of events that could have occurred, then the part of this whole, where the first of the 4 people purchased an item, is 2/3 of the whole. Then the probability of finding a second person from the remaining crowd that also bought at least one item is 79 (because we already found one) out of 119 (again, because we have one less person to choose from in the crowd). The probability for the third person is found in a similar fashion...it would be 78/118 and 77/117 for the fourth person. We are each time finding an even smaller fraction of the whole that represents the final probability. If the probability for each was the same (independent events) then we could see that the area was decreasing exponentially (something I was never explicitly taught in school). To find the probability of all 4 people having bought at least 1 item, you need to find the product of the probabilities for each person. The answer then is 2/3 x 79/119 x 78/118 x 77/117 = approximately 19.25%.

TomCat said...

Alien Citizen said...

:-) Thanks. This is what I do for fun...and, sometimes, what I teach.

TomCat said...

I taught myself trig and calculus for fun, but I've forgotten a lot of it. What surprised me was how calculus reached a point where left-brain solutions failed and I had to visualize the solution before I could integrate the equation.

Alien Citizen said...

Wow, Tomcat. I love to meet more people who appreciate the beauty of math. In fact, I taught myself all of the high school math subjects because I wasn't in school when I was a teen (I was living abroad with family). Occasionally, there is some topic I find that others are familiar with and I didn't even realize I was missing but I did successfully earn my math degree so I'm happy.

Yes! Integral Calculus is a real surprise in that way. I just wanted my professors to tell me how to solve it mechanically but you're right...can't really be done. Very interesting and contrary to the general view of what math is. Thanks for the comment. :-)