Elementary probabilities question

While I am good with numbers and general math, I have never taken a stats and probabilities class. What little I have learned has all been gleaned from the good folks here on this forum.
So it has now come to test time for me. I would like to know if I am calculating these two items correctly.

Problem 1. What is the chance I can roll three 4s, before a 7 is rolled?

3 ways to roll a 4, 6 ways to roll a 7, for a 1:2 ratio --funny how that matches the free odds, ;-)

so 50% chance that I would get a 4 before a 7, correct?

1/2 x 1/2 = 1/4, so 25% chance I will roll two 4s before a 7. Is this correct?

1/2 x 1/2 x 1/2 = 1/8 = 12.5% chance of getting three 4s before a 7?


Problem 2. What are the chances of rolling any place number (4,5,6,8,9,10) before a 7?

24 ways of rolling a place number, 6 ways to roll a 7.
So 1 out of 4 rolls is a 7, meaning 3 out of 4 rolls will be a place number, means 75% chance of rolling a place number before a 7?

And rolling 2 place numbers before a 7 would be 3/4 x 3/4 = 9/16 = 56.25% ?

so rolling 3 place numbers before a 7 would be 3/4 x 3/4 x 3/4 = 27/64 = 42.19% ?

and finally, rolling 4 place numbers (which is the slight profit point if you place all 6 numbers across) would be
3/4 x 3/4 x 3/4 x 3/4 = 81/256 = 31.64%

Have I done this correctly ?

1/3 chance to roll 4 before 7. 1/27 to do it 3 times.

The concept is also called the craps principle
http://en.wikipedia.org/wiki/Craps_principle

For at least one time:
Place bets 24 ways
7 has 6 ways

24 / 24 + 6 = 24/30 or 4/5 = 0.8 *100 = 80%
or (24/36) / (24/36)+(6/36)

The 4 is done the same way
3 / 3 + 6 = 3/9 or 1/3

For at least N times = p^N

Any place bet table
80% chance of getting at least 1 place number before a 7
16% chance of getting exactly one before a 7.

X	at least X	exactly X
1	80.00%	16.00%
2	64.00%	12.80%
3	51.20%	10.24%
4	40.96%	8.19%
5	32.77%	6.55%
6	26.21%	5.24%
7	20.97%	4.19%
8	16.78%	3.36%
9	13.42%	2.68%
10	10.74%	2.15%

One has a 51.2% chance of getting at least 3 place numbers before one 7
or a 48.8% chance of getting 2 or less (at most) place numbers before one 7

A Wizard's example can be found in this thread
https://wizardofvegas.com/forum/questions-and-answers/math/764-long-odds/2/#post3397

Sally

Thanks for the explanation. I see now where I went wrong.

Not picking on RaleighCraps...

But, this is a good example of using a ratio (odds) to find a probability

Quote: RaleighCraps

Problem 1. What is the chance I can roll three 4s, before a 7 is rolled?

3 ways to roll a 4, 6 ways to roll a 7, for a 1:2 ratio --funny how that matches the free odds, ;-)

so 50% chance that I would get a 4 before a 7, correct?

Your 1:2 ratio is correct (3:6)
The ratio is just another way to show the odds for an event and not the probability.

the Odds do not equal the probability of an event

But from the odds we can find the probability.

1:2 means 1 chance to win and 2 chances to lose. (or 3 win : 6 lose)
(true payoff odds like 2 to 1 (2:1) means 2 ways to lose and 1 way to win)

since probability considers all the possible outcomes of an event
1+2=3 total possible outcomes
only one way to win
1/3 is the probability

Probability formula =
(ways to win / total possible outcomes)

This matched the title of your thread.

Quote: RaleighCraps

Problem 2. What are the chances of rolling any place number (4,5,6,8,9,10) before a 7?

24 ways of rolling a place number, 6 ways to roll a 7.

I agree. We are working with ratios.
24 ways to win
6 ways to lose for a 4:1 ratio (24:6)
4+1=5 total possible outcomes (24+6)
4 ways to win
4/5 is the probability



Now up to the reader to answer the next questions.

Quote: RaleighCraps

So 1 out of 4 rolls is a 7,

Is "1 out of 4" the same as the ratio of 1:4 (from 6:24)???

Quote: RaleighCraps

meaning 3 out of 4 rolls will be a place number, means 75% chance of rolling a place number before a 7?

Great job
Ice cream for everyone


It is very easy to mix up ratios and probabilities.
(I catch myself doing it quite often)
They do not mean the same thing.
From a ratio we can find the probability.

Enjoy

Quote: guido111

Not picking on RaleighCraps...

It is very easy to mix up ratios and probabilities.
(I catch myself doing it quite often)
They do not mean the same thing.
From a ratio we can find the probability.

Enjoy

Personal attack. Suspension. Nah. j/k

No offense taken whatsoever. That was a very good read on the difference between a ratio and probabilities.

I'm glad that was not a test, as I really couldn't even figure out a way to ask for partial credit.... :-(

At least I was able to correctly figure out the number of ways a number could be thrown.

Quote: RaleighCraps

Personal attack. Suspension. Nah. j/k

No offense taken whatsoever. That was a very good read on the difference between a ratio and probabilities.

I'm glad that was not a test, as I really couldn't even figure out a way to ask for partial credit.... :-(

At least I was able to correctly figure out the number of ways a number could be thrown.

And by knowing a little bit of truth, the chances of a number rolling, you actually can end up with other accurate values using different methods to calculate them.
That is what I like about math.

Many times, many different roads will all lead to the correct answer.
Hey, thanks for the excellent questions!

Very similar to the "Four rolls before a 7" problem that came up in the last month or so.
BTW: "Four before 7" pays even money with a 3.55% House Advantage.