Probability of not getting a prize

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In summary, the conversation discusses the probability of winning a prize when purchasing two tickets from a pool of 10,000 tickets, where there are only 10 winning tickets. It is determined that the probability of not winning is equal to the probability of not winning on the first ticket multiplied by the probability of not winning on the second ticket, which is equal to 1/2. This is also equivalent to the probability of not winning being equal to 1 minus the probability of winning at least one ticket, which is also 1/2. There is also discussion about the original approach used, which did not account for the multiple ways to win on two tickets, and the revised approach that does take this into account.
  • #36
haruspex said:
It depends whether order matters. In the present case, it makes no difference whether you pick a winner, then a loser, or the other way around. So for k selections, the factor k! appears in the denominator for both the number of successful combinations and the total number of combinations.

In a random walk with an absorbing barrier the order does matter. If one step away from the barrier, what is the probability of reaching it in three goes (with 50:50 at each step)?
You only need two of the three in the right direction: an evens chance.
But if the first step is in the right direction it doesn’t matter about the other two: so 5/8.
I guess I didn't realize that might not be the case for all classes of problems. I don't have any formal coursework in the subject.
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  • #37
erobz said:
If you are going to reach into the bag pull three marbles, what the probability of getting 2 red 1 green marble?

Using Combinations:

$$ P(2r,1g) = \frac{C(4,2)C(2,1)}{C(6,3)} = \frac{3}{5}$$

Using Permutations:

$$ P(2r,1g) = \frac{ 4 \cdot 3 \cdot C(3,2) \cdot 2}{ 6 \cdot 5 \cdot 4} = \frac{3}{5}$$

Combinations are computationally superior...I guess. Both work.
As pointed out by @haruspex, when probability is determined, a common factor appears in numerator as well as denominator when taking permutations in this example according to the identity below. That's why using either permutations or combinations to determine probability gives the same value in the example of selecting marbles.

$${}^nP_r =r! \times {}^nC_r $$
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