I was posed the following question recently:
Suppose a brand of cereal has four different toys included in their cereal boxes, each with an equal probability of being found: On average, how many boxes of cereal do you need to buy to get all four toys?
Since getting a toy in each box is an independent event and the probability stays constant, I guessed that you could model this using the binomial distribution:
\begin{align} X &= \text{number of unique toys}\\ X &\sim B(n, 0.25) \end{align}
Where $n$ is the number of trials. The probability function for getting four unique toys would therefore be:
$$P(X = 4) = {n \choose 4} \cdot 0.25^4 \cdot 0.75^{n-4}$$
So, to find the mean number of trials, I perform the infinite sum:
$$\sum \limits_{n=4}^\infty \left[n\cdot P(X=4)\right] = \sum \limits_{n=4}^\infty \left[n\cdot {n \choose 4} \cdot 0.25^4 \cdot 0.75^{n-4}\right]$$
Which, when I plug this into Wolfram Alpha returns $76$.
So, according to my calculation, on average you need to buy $76$ boxes of cereal to get all four toys. Is this right?
