Another way to define probabilities is under Bayesian perspective. Bayesian perspective is one of personal perspective. Your probability represents your own perspective, it's your measure of uncertainty, and it takes into account what you know about a particular problem. But in this case, it's your personal one, and so what you see, may be different than what somebody else believes. For example you want to ask, is this is a fair die? If you have different information than somebody else then your probability of the die is fair may be different than that persons probability. So inherently a subjective approach to probability, but it can work well in a mathematically rigorous foundation, and it leads to much more intuitive results in many cases than the Frequentist approach. We can quantify probabilities by thinking about what is a fair bet. So for example, we want to ask what's the probability it rains tomorrow? Then we can ask about a bet that you might be willing to take if you thinks it's fair. Suppose you'd be willing to take the bet that if it rains tomorrow, you win $4. If it doesn't rain tomorrow, you lose $1, or whatever your local currency is. You can think of this as odds of 4 to 1. If you think that bet's, fair bet you should be willing to take it in either direction which would also mean if it rains, you lose $4. And if no rain, you win a $1. If you think these are both fair, then you're defining a probability of rain as 1 over 1 plus 4 which equals 1 and 5. We can see this is fair by looking at your expected return. Under the first bet, your expected return is you win 4 with probability 1 and 5 and you lose 1 with probability 4 and 5. And you can see, that that is a fair bet of 0. In the second case, you win 1 with probability of 4 and 5, and you lose 4 with probability 1 and 5, and that also is 0. So it balances out. It only balances out if your probabilities match. So you can use this betting framework to think about what is your personal probability based on what bets you would be willing to take. In most cases, it's fairly easy to bracket it. You take a bet that was a thousand and one in favor, but you wouldn't take a bet that was a thousand and one against. And so your personal probability is somewhere in between. Finally, I want to mention the concept of coherence. Probabilities must follow all the standard rules of probability, those that were defined in the supplementary material for this lecture. If you don't follow all the rules for probability, then you can be incoherent, which leads to a case for someone to construct a series of bets where you're guaranteed to lose money. This is referred to as Dutch book. If you do follow all the rules, and you follow the framework of Bayesian statistics, then you can be guaranteed to be coherent.