There are 12 distinct prizes and we can use a ticket to draw one of them of those 12 with equal probabilities. I have 7 of those prizes already and I have 7 tickets. What is the expected number of new prizes that I can get?
Let's define a random variable X as the number of new prizes you will get after using all 7 tickets. The possible values of X are 0, 1, 2, 3, 4, 5, 6, and 7.
To calculate the expected value of X, we can use the formula:
E(X) = Σ(x * P(X=x))
where x is the possible value of X, and P(X=x) is the probability of getting x new prizes.
If you have already won 7 prizes, then there are 5 remaining prizes that you can potentially win with your 7 tickets. Since each ticket gives you an equal chance of winning any of the 5 remaining prizes, the probability of winning a particular new prize with one ticket is 1/5.
Therefore, the probability of getting x new prizes out of 7 tickets is:
P(X=x) = (5 choose x) * (1/5)^x * (4/5)^(7-x)
where (5 choose x) is the binomial coefficient that gives the number of ways to choose x prizes out of 5 remaining prizes.
Now we can calculate the expected value of X:
E(X) = Σ(x * P(X=x))
= Σ(x * (5 choose x) * (1/5)^x * (4/5)^(7-x))
≈ 1.941
So the expected number of new prizes you can get with your 7 tickets is approximately 1.941