Probability based on position

Question

I am trying to find the probability of two events occuring one after the other based on their positions. For instance,if there are $25$ elements with equal probability of choosing them what is the probability of the first element being chosen and the second element is chosen, likewise it may be second and third or third and fourth, fourth fifth etc? Will the following help: $$\frac{1}{25}\cdot\frac{50}{100}\cdot\frac{1}{24}\cdot\frac{50}{100}$$ But I guess this one cannot give me the position of the event occurence.

Answer 1

This is known as the probability of Dependent events. This means:

If two events $A$ and $B$ are dependent, then the probability that $A$ and $B$ will occur is

$P(A$ and $B) = P(A)*P(B|A)$

For instance, if you are asking the probability of the first two elements being chosen one after the other, you obtain

$P(A$ and $B) = \frac{1}{25}*\frac{1}{24} = \frac{1}{600}$

This is because the first element is not being replaced after it is drawn, so the total number of elements decreases by 1 each time you draw an element.

I hope this can help.