Chp 2 Notes: A Gentle Introduction Flashcards
What is expected value?
The expected value or expectation is the long term average of a large number of independent experiments
What is the law of large numbers?
According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.
What are event trees?
Event trees are simple figures that can aid in reasoning about probability problems. The root of the tree represents the beginning of the experiment. The children of the root represent the first step of the experiment, the children of these children represent the second step of the experiment, etc. Each edge is associated with a probability, and the probabilities of all edges emanating from a node sum to one.
What is uniform distribution?
When all outcomes have equal probability, we say that the distribution over the outcomes is uniform.
The three cards problem: The house has three cards: one card is red on both sides, one card is black on both sides, and one card is black on one side and red on the other side. We denote the three cards by the letter pairs: BB, BR, RB.
The house choses one of the cards at random and puts it down on the table with a randomly chosen side facing up. Suppose the face up side is black.
The house makes the following offer: if the other side of the card is red, then the house will give you a dollar, if the other side is black, you give the house a dollar.
If the side facing up is red, then the reverse offer is made: if the other side is black, the house will give you a dollar, otherwise you give the house a dollar.
The house claims the game is fair - say the face up side is black, then there are two possible cards: BB and RB. As both of these cards are equally likely to be chosen, the probabilities that the other side is black or red are both 1/2 and the game is fair.
Is the game fair?
We need to carefully define the event tree and the outcome space. To aid the explanation, study the event tree in Figure 2.3. The event tree has two levels, the first corresponding to the choice of card, the second to the choice of card side.
As a result, we get six possible outcomes, each with probability 1/6. Note that there are two outcomes corresponding to the two sides of each card, depending on which side of the card is facing up, as denoted by the up arrow: ∧. This is obvious for the card RB. It might be less obvious for the cards RR and BB, because both sides of the card have the same color. However, the two sides are obviously distinct, just imagine that we marked one side of each card using invisible ink.
The rest of the argument is simple. Consider Figure 2.3 again. Suppose the face up side of the chosen card is red. There are three outcomes that correspond to this event. Restricting our attention to these three outcomes, we see that two correspond to the card RR and one corresponds to the card RB. Thus the probability that the bottom side is red “conditioned” on the fact that the top card is red is 2/3. This type of probability is called “conditional” probability.
