Probability of 2 equivalent random selections from integer sets

[Credulous]

What is the probability that a number selected from 0-9 will be the same number as one randomly selected from 0-4?
Relevant equations: $$P(A \cap B) = P(A)*P(B|A)$$
I used the equation above, using A as the event that the number selected from 0-9 will be between 0 and 4, and B as the event that the two selections are the same. Putting these two together I got: $$P(A)*P(B|A) = \frac{5}{10}*\frac{1}{5}^2 = 1/50$$.

It seems alright but it feels too small of a chance for this to happen. I don't really understand probability theory that well. Any books to recommend?

 

[tiny-tim]

Welcome to PF!

Hi Credulous! Welcome to PF!

$$P(A)*P(B|A) = \frac{5}{10}*\frac{1}{5}^2 = 1/50$$.

Why squared?

 

[Ray Vickson]

What is the probability that a number selected from 0-9 will be the same number as one randomly selected from 0-4?



Relevant equations: $$P(A \cap B) = P(A)*P(B|A)$$



I used the equation above, using A as the event that the number selected from 0-9 will be between 0 and 4, and B as the event that the two selections are the same. Putting these two together I got: $$P(A)*P(B|A) = \frac{5}{10}*\frac{1}{5}^2 = 1/50$$.

It seems alright but it feels too small of a chance for this to happen. I don't really understand probability theory that well. Any books to recommend?

Sometimes (not always) the most enlightening way to solve a probability problem is to construct the actual "sample space" and look at the event you are interested in. In this case, the sample space consists of all pairs of the form (a,b), where 0 <= a <= 4 and 0 <= b <= 9 are integers. If E is the event "equal numbers", what is E, as a subset of the sample space? What is the probability p(a,b) of sample point (a,b)? How would you get the probability of E?

RGV