Question

A die is rolled 20 times and the number of twos that come up is tallied. Find the probability of getting exactly four twos.
A. 0.202
B. 0.075
C. 0.101
D. 0.083

Answer 1

This is a pretty standard probability question.
What is the P of getting 4 twos in a row, followed by 16 rolls with no twos?

Answer 2

And then, what is 20 choose 4?

Answer 3

choose ? x

Answer 4

How many ways are there to choose 4 items from a set of 20. These are called combinations.

Answer 5

how do i make a formilu

Answer 6

I'm getting there. The formula has two parts. The probability in my first post above times the value of what's called 20 choose 4, the number of combinations of 4 items from 20 total. The first term is \[(1/6)^{4} (5/6)^{16}\] The second term is 20 x 19 x 18 x 17 / 4!.

Answer 7

Can you do that multiplication?

Answer 8

20x19=380x18=6840x17=116280/4=29070

Answer 9

Yes, for the second term. How about the first?

Answer 10

Actually, I get a different answer for the second term. Let Python do the work:

>>> a = (1.0/6)**4
>>> a
0.0007716049382716048
>>> b = (5.0/6)**16
>>> b
0.05408789293239543
>>> c = 20 * 19 * 18 * 17 / 24.0
>>> c
4845.0
>>> a*b*c
0.2022035812171727

Answer 11

Read the first part of this: http://en.wikipedia.org/wiki/Binomial_distribution

Answer 12

that looks krazy

Answer 13

Well, OK. But to say it again, the probability of 4 twos in a row followed 16 rolls with no twos is: \[(1/6)^{4} (5/6)^{16}\]
You agree with that?

Answer 14

yes

Answer 15

but how did you get this

>>> a = (1.0/6)**4 >>> a 0.0007716049382716048 >>> b = (5.0/6)**16 >>> b 0.05408789293239543 >>> c = 20 * 19 * 18 * 17 / 24.0 >>> c 4845.0 >>> a*b*c 0.2022035812171727

Answer 16

Maybe I shouldn't have posted that. It's a programming language called Python and I use it to do moderately complex calculations without making mistakes. Notice that I obtained one of your possible answers as my answer.

Answer 17

The trick to the calculation is to realize that although the probability of getting your 4 twos right at the beginning is very low (about 4.17 x 10^-5.. in the problem statement they can come at any point during the series. That's a combinations problem. And the answer they show is the classic formula.

Answer 18

The second "they" being the wikipedia article.

Answer 19

is this one the same
If 5 apples in a barrel of 25 apples are rotten, what is the expected number of rotten apples in a sample of 2 apples?
A. 0.4
B. 1
C. 0.63
D. 0.33

Answer 20

Anyway, that's how you do this kind of problem. Can you guess for a series of n rolls what is the most likely value for the number of twos? Hint: it's what you would naturally guess.

Answer 21

Umm.. The P of getting a bad one one for the first pick is 1/5, and almost 1/5 for the second one (depending on what happened the first time). The expected value is the sum of these, I think… Anyone else?

Answer 22

The answer is found from the binomial distribution as follows:
\[P(4twos)=\left(\begin{matrix}20 \\ 4\end{matrix}\right)(\frac{1}{6})^{4}(1-\frac{5}{6})^{16}=0.2022\]

Answer 23

I think you have a typo

Answer 24

And for @85295james, that funny thing in the parentheses at the beginning of his formula is the official way to say "20 choose 4"

Answer 25

o

Answer 26

You are right.
\[P(4twos)=\left(\begin{matrix}20 \\ 4\end{matrix}\right)(\frac{1}{6})^{4}(1-\frac{1}{6})^{16}=0.2022\]

Answer 27

Yes, that's good.

Answer 28

so the p of getin two rotten apples in a row is high but how do i put that in math

Answer 29

\[\left(\begin{matrix}20 \\4\end{matrix}\right)=\frac{20!}{4!16!}\]

Answer 30

Which is the same as 20 x 19 x 18 x 17 / 4 x 3 x 2

Answer 31

Well, for your apples problem, it is just about 1/5 plus 1/5 so I'd go with 0.4.

Answer 32

116280 / 24 =486.66

Answer 33

Do the division again..

Answer 34

this is "/" divide

Answer 35

116280 divided by 24 is equal to 4845

Answer 36

o yes it is

Answer 37

See ya. Think about my bonus problem.

Answer 38

ok

Answer 39

but how about the apples

Answer 40

See above, about 10 posts up.

Answer 41

1/5 + 1/5 =0.4 how did you get that