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Note to MATH 105 instructors: here my goal is to define discrete probability with an example"

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 pk to denote the probability that event k will occur.

Discrete Probability Rules
In discrete probability,
1. Probabilities are numbers between 0 and 1: 0 ≤ pk ≤ 1 for all k
2. The sum of all probabilities for a given experiment is equal to one: ∑k p = 1

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

• p1 to be the probability that the tossed coin will land as heads
• p2 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.

• p1 = 0.5
• p2 = 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

• p1 to be the probability that the outcome will be HH
• p2 to be the probability that the outcome will be HT
• p3 to be the probability that the outcome will be TH
• p4 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, p1 = p2 =p3 =p4 = 0.25.

Again, all of our probabilities sum to 1, and each probability is a number on the interval [0, 1].