What are the odds question

Question

There are 25 balls numbered 1 thru 25 in a hat. I have numbers 1, 2, & 3. What are the odds that the last 3 numbers drawn are mine, 1, 2, & 3? Once a number is drawn, it is not put back into the hat. What is the answer and how do you solve this probability question?

Answer 1

Hi Ralphie P.,

This is a good question to use to illustrate the value of thinking a problem through before calculating.

If you took the brute force approach, and started your calculation with the probability of drawing any of the other (initially 22) other balls, etc. you would have a string of calcs:

P = (22/25)*(21/24)*(20/23).....

Now you could start doing each of these quotients, and then multiplying ...

What a waste of time! First of all, the numerator and denominator series have substantial overlap, allowing cancellations.

But ALSO (and mathematically equivalently), there is no reason to suspect that (1,2,3) will be variously drawn first, or last, is there?

So reframe the question as drawing those 3 numbers (you don't indicate if they have to be drawn in order, I'll assume not) first:

that's (3/25)*(2/24)*(1/23) -- done. The logic here: 3 possibilities "correct" for drawing the first ball out of 25, 2 possibilites of drawing another "correct" ball from the remaining 24, and only 1 possibility for drawing the last "correct" ball from the remaining 23. I'll let you do the math.

--Cheers, -- Mr. d.