Combination question - Average before same combination repeats

[Ajax]

Hi guys, hoping someone can help me with this. I’ve posted this on a couple other forums and so far I haven't had any kind of solid answer on it! I was always better with words than numbers so I have no idea where to begin with this.

So imagine a combination lock with four values in length. The first value is from a choice of two numbers, the second from a choice of two numbers, the third from a choice of four numbers and the fourth from a choice of four numbers.

So it would look like this

?-?-?-? <-- possible combination made
1-1-1-1
2-2-2-2
x-x-3-3
x-x-4-4
--------
a-b-c-dIf a combination was randomly generated, I’m trying to work out on average how many combinations you’d likely go through before one matched the first if 70% of the time the value in column A was the number 1 and 80% of the time the value in column B was the number 1 (the values in column C & D each have a 25% chance). So the odds percentage wise are;

70-80-25-25
30-20-25-25
00-00-25-25
00-00-25-25

I'm also interested in how to work that out so I can change values and percentages and come up with likely averages myself depending on what criteria I’m needing at the time.

Can anyone help and explain how it was worked out so I can learn for myself?

 

[Aaron Bex]

Thanks!

The total number of possible combinations for the lock is 2 x 2 x 4 x 4 = 32. The probability of a randomly generated combination matching the first if 70% of the time the value in column A is the number 1 and 80% of the time the value in column B is the number 1 is (0.7 x 0.8 x 0.25 x 0.25) = 0.015625. This means that on average, you would have to go through 1/0.015625 = 64 combinations before you find one that matches the first.

If you want to be able to change the values and percentages yourself, the formula is:

1 / (probability of each value in columns A & B x probability of each value in columns C & D) = Average number of combinations before finding a match