Question

The chance of a day being rainy is 20%. What is the chance of there being two rainy days during a week?

Answer 1

Start by finding out how many rainy day combos there are in a week. Then, take that number and multiply it by (%rainy)^2 and multiply it again by (%not rainy)^5.

Answer 2

Wouldn't the rainy day combos be 7^7?

Answer 3

There is an easy way to figure the combos. It's starts by finding the permutations first. I'll explain.

Answer 4

Look at the days, Sunday thru Saturday, as days 1-7. You could have the any of the days 1-7 as the "first" rainy day, so you start with a factor of 7.

Answer 5

Now, ther are six days left for the "second" day (which may or may not precede the "first" day, hence the apostrophe's).

Answer 6

So, there are 6 days from which to choose from for the "second", so the next step is 7 x 6, but you don't want to double count the pairs, so you have to divide by 2, so 7 x 6 / 2.

Answer 7

7 x 6 / 2 is the number of combinations of 2 rainy day combos. There are a couple of other ways to get to this number. Perhaps the easiest is to start with day 1 and realize that there are 6 other days it can be paired with "going forward". Then take day 2 and realize that there are 5 to be paired with, etc. 6 + 5 + 4 + 3 + 2 + 1 = 42

Answer 8

So 21 days that have 2 rainy days?

Answer 9

yes

Answer 10

Not 21 days have rainy days, but more accurately, there are 21 pairs of days that are rainy.

Answer 11

But how many total combinations are there

Answer 12

7! or 5040?

Answer 13

That's step1. Now, the second part is covered in the first reply. Multiplying by the probability of rainys x non-rainys

Answer 14

What you are really asking is how many total pairs of days are there in 7 objects (days, dice, balls, hoops, etc.). Just 21 ways to take 2 differing pairs of things from a set of 7 different things.

Answer 15

So about .275

Answer 16

Right?

Answer 17

yes

Answer 18

K, Thanks

Answer 19

You got it right and very quickly I might add

Answer 20

Would the process be similar if the question was find the probability of at least 2 rainy days?

Answer 21

Then, you would have to add up 2 rainy days + 3 rainy days + 4 +... Easier would to be finding the probability of no rainy days + 1 rainy day and subtracting that from 1.

Answer 22

the probability for P(rain on 0 days) = .209 and the the probability for P(rain on 1 day) = .367 so the answer would be 1-.209-.367 which equals .424, right?

Answer 23

Yes, that's right. You're actually pretty good at probabilty. I don't think you are going to have much trouble with this subject.

Answer 24

I say that because you got both of these answers very quickly with minimal help.

Answer 25

Mind answering another?

Answer 26

Ok, but I don't think I'm as fast as you are!

Answer 27

Here goes: First flip a coin. If you got heads, roll a 6 sided dice. If you got tails, roll an 8 sided dice. What is the chance that you flipped heads if you rolled a 6?

Answer 28

I can do this one, but you'll have to bear with my step-by-step.

Answer 29

K, go for

Answer 30

Start with a "backwards to forwards" mindset on this. That is what you have to do with what is essentially an example of Bayes' Theorem. (Uses conditional probability). If you roll a 6-sided die, the probability is 1/6 or 4/24. If you roll an 8-sided die, the prob is 1/8 or 3/24. The total probability is 7/24. So, 3/24 / 7/24 = 3/7. Now let me think about this to see if my logic was right.

Answer 31

Got my 6 and 8 switched. 4/24 / 7/24 = 4/7.

Answer 32

I could draw you a nice matrix for this. How do I use a drawing tool here?

Answer 33

so would the answer br 4/7 or .571?

Answer 34

I just saw the drawing tool button

Answer 35

Yep, it's right. Thanks so much

Answer 36

Yes, that's the answer, but you might want to spend some time with this and the drawing as it is essentially a harder problem than the first. I'm going to try to draw a matrix. It's my first one, so it will be sloppy.

Answer 37

Would you mind answering another question at about 7:00 PST, I have a piano lesson now.

Answer 38

If i'm around, just look for me.

Answer 39

I already became a fan, so that should be easy.

Answer 40

Thank you very much

Answer 41

good luck, Chopin.

Answer 42

So it is really (4/48) / (7/48) = 4/7 which is the same answer of 0.571. With this drawing, I just divided the numerator and denominator by 1/2 to clarify the true probabilities of getting a "6". Same answer, just more rigorous.