# How to solve this probability problem analytically (instead of using simulation)? Probability of winning prize

I ran across a probability problem as follows:

In a certain prize give away, to win a prize, you must spell the word "WIN". The tickets are printed so that 70% of them have a "W", 20% have an "I", and 10% have an "N". If you purchase 10 tickets, what is the probability that you will win the prize?

I have an estimate of 0.6 (if rounding to one decimal place) through simulation, but I don't know how to solve this problem analytically and what probability distribution should be used.

• en.wikipedia.org/wiki/Multinomial_distribution Each ticket gives you a W, I, or N with probability .7, .2, and .1. What's the probability that 10 tickets contains at least one of each? Feb 14, 2020 at 9:55
• Thank you @assumednormal. My struggle with using multinomial distribution directly lies in the large amount of possible combinations for $n_{W}$, $n_{I}$, and $n_{N}$ for fully describing "10 tickets contains at least one of each". Would you mind telling me a bit more about the multinomial-based solution? Thanks! Feb 14, 2020 at 14:09
• You can view all these combinations on Wolfram alpha. Then, to obtain the probability, just set all the variables to $1$ (redo the calculation with /. {W->1,J->1,N->1} appended). This sort of calculation can be instructive.
– whuber
Feb 14, 2020 at 17:30

## 1 Answer

Let's find the case we couldn't WIN. Call the individual probabilities as $$p_w,p_i,p_n$$, and call it $$k$$ trials: $$P(\text{WI})=(1-p_n)^{k}, P(\text{WN})=(1-p_i)^{k}, P(\text{NI})=(1-p_w)^{k}$$ $$P(\text{W})=p_w^{k}, P(\text{N})=p_n^{k}, P(\text{I})=p_i^{k}$$

Via Inclusion-Exclusion Principle:

$$P(\text{WIN})=1-(P(\text{WI})+P(\text{WN})+P(\text{NI}) - P(\text{W})-P(\text{N})-P(\text{I}))\approx 0.572$$