Probability Question

[hard_hook]

If I have 74 numbers (1-74) and I select 40 what are the odds/probability that I would be able to guess 24 out of those 40 numbers?

And what are the odds/probability that I could guess the first 24 in a row?

I know.,....I can't wrap my mind around how to do this!

 

[soroban]

Hello, hard_hook!

Could you state the problem again?
As given, it doesn't make sense.

If I have 74 numbers (1 - 74) and I select 40,
what is probability that I would be able to guess 24 out of those 40 numbers?

Since you selected the 40 numbers, you know what they are, right?
. . Then there isn't any "guessing".

 

[hard_hook]

Sorry for the confusion

I have 74 numbers. Before doing anything I "guess" and write down 24 numbers.

Now I start pulling out numbers from the 74. What is the probability that I would get all my 24 numbers (That I wrote down prior) after selecting 40 numbers from the 74.

And second....what is the probability that once I start pulling out numbers from the 74 that I would select the 24 I guessed in a row.

Does that make sense?

 

[pka]

Now I start pulling out numbers from the 74. What is the probability that I would get all my 24 numbers (That I wrote down prior) after selecting 40 numbers from the 74.

There are \(\displaystyle \binom{74}{40}\) subsets of forty from a set of seventy-four.
Of those there are \(\displaystyle \binom{50}{16}\) subsets that contain any specified set of twenty-four.
So divide the second number by the first.

As for the second question, things are still unclear.
I understand that the twenty-four are ordered.
It seems that you asking about drawing those twenty-four in the given order. That much is fine.
However, do you mean that some string of twenty-four must match the original? OR simply, all twenty-four must appear somewhere in the string of forty in the original order.

 

[soroban]

Hello, hard_hook!

I'm still not sure of the instructions.
I will make my own assumptions about what is intended.

I have 74 numbers. . I choose and write down 24 numbers.

Now I pull out 40 numbers from the 74.
What is the probability that I would get all my 24 numbers?

There are \(\displaystyle {74\choose40}\) ways to draw 40 numbers.

There are 24 Chosen numbers and 50 Others.

We want to draw: 24 Chosen and 16 Others.
. . There is 1 way to get the 24 Chosen.
. . There are \(\displaystyle {50\choose16}\) ways to get 16 Others.

The probability is: .\(\displaystyle \dfrac{1\cdot{50\choose16}}{{74\choose40}}\)

What is the probability that, once I start pulling out numbers from the 74,
I would select the 24 I guessed in a row.

I assume this means: you draw out all 74 numbers
and you want your 24 Chosens to be in consecutive draws in their original order.

There are 74! possible outcomes of the 74 draws.

There is one way for the 24 Chosens to be consecutive and in order.
This string can occupy 50 possible positions, from 1-to-24 to 50-to-74.
The 50 Others can be placed in 50! possible orders.

The probability is: .\(\displaystyle \dfrac{1\cdot 50 \cdot 50!}{74!}\)