Science:MATH105 Probability/Lesson 1 DRV/1.01 Discrete Random Variables
Note to MATH 105 instructors: here my goal is to define discrete probability with an example"
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Probability
In many areas of science we are interested in quantifying the probability that a certain outcome in an experiment occurs. To quantify the probability that an event occurs, we use a number between 0 and 1 that represents how likely that event occurs. A probability of 0 implies that the outcome cannot occur, whereas a probability of 1 implies that the outcome must occur. Any value in the interval (0, 1) means that the outcome only occurs some of the time.
Rules of Probability
When there is a discrete list of probabilities that can occur, we use the notation p_{k} to denote the probability that event k will occur.
Discrete Probability Rules 

In discrete probability,

Example: Tossing a Fair Coin Once
If we toss a coin into the air, there are only two possible outcomes: it will land as either "heads" (H) or "tails" (T). If the tossed coin is a "fair" coin, it is equally likely that the coin will land as tails or heads. In other words, there is a 50% chance that the coin will land heads, and a 50% chance that the coin will land as tails.
Using our notation for probability, we can assign
 p_{1} to be the probability that the tossed coin will land as heads
 p_{2} to be the probability that the tossed coin will land as tails
Because there are two outcomes that are equally likely, we assign the probability of 0.5 to each of them.
 p_{1} = 0.5
 p_{2} = 0.5
As required, the sum of the probabilities equals 1, and each probability is a number in the interval [0, 1].
Example: Tossing a Fair Coin Twice
Similarly, if we toss a fair coin two times, there are four possible outcomes. Each outcome is a sequence of heads (H) or tails (T):
 HH
 HT
 TH
 TT
Using our notation for probability, we can assign
 p_{1} to be the probability that the outcome will be HH
 p_{2} to be the probability that the outcome will be HT
 p_{3} to be the probability that the outcome will be TH
 p_{4} to be the probability that the outcome will be TT
Because the coin is fair, each outcome is equally likely to occur. There are 4 possible outcomes, so we assign each outcome a probability of 1/4 = 0.25. That is, p_{1} = p_{2} =p_{3} =p_{4} = 0.25.
Again, all of our probabilities sum to 1, and each probability is a number on the interval [0, 1].