combinatorics

[summer101.blues]

Hi,

I saw this video by Ron on combinatorics and was very impressed by his dash method to solve combinatorics problems. However i struggled with a question when i try to solve a problem using that approach.
This problem i have made up so no source as such though i have seen similar problems
--> Say chances of rain is 20% on any given day. calculate the probability that it will rain on exactly 3 days out of 4.

Can i solve this using dash method?

[tim]

Before we help with this question, we need you to show some effort of your own. What did you try on this question? Where did you get stuck?

[summer101.blues]

OK, the probability of rain on only 3 days out of 4 = (0.2)(0.2)(0.2)(0.8) or (1/5)(1/5)(1/5)(4/5)
now we multiply 4!/3!1! to get the answer.

but in the Ron's combinatorics tutorial, we should just divide by 3!

can you explain the disconnect?

[tim]

please tell me more about what you mean by "just divide by 3!"

as opposed to what?

[summer101.blues]

Hello Tim,
Sorry i might sound confused but questions regarding probability and combinatorics befuddle me.

Slot method steps when order does matters were
a. Create slots for each decision that must be made
b. Fill the slots with the number of options
c. Multiply the number

In the problem stated why can't i use the above method by just multiplying probabilities (1/5)(1/5)(1/5)(4/5)?

[tim]

okay, this is very different from your "just divide by 3!" question, but i think i can help here. anytime you calculate a probability, you must ask yourself what are you calculating - get the numbers to tell a story. the 1/5 to start means it rained on the first day. same thing for days 2 and 3. the 4/5 at the end means it didn't rain on the fourth day. that's fine if this is what you wanted to calculate, but since you wanted to account for all the possibilities you have to multiply by 4. of course, you could have computed all four narratives separately:

1/5 1/5 1/5 4/5
+
1/5 1/5 4/5 1/5
+
1/5 4/5 1/5 1/5
+
4/5 1/5 1/5 1/5

but as you can see, this is just four times the first one! :)