# 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"

## 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. There are two conventions that we use:

Discrete Probability Rules
In discrete probability,
1. Each probability is a number between 0 and 1: 0 ≤ pk ≤ 1 for all k
2. The sum of all the probabilities 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:

• HH
• HT
• TH
• 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.

## Discrete vs Continuous Random Variables

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In MATH 105, we deal with

• discrete random variables, and
• continuous random variables.

We will briefly explore discrete random variables first. This will be in order to introduce the notions of probability, mean and variance, that we will use when discussing continuous random variables.

These concepts are introduced assuming no prior knowledge of probability, but they are discussed with an assumption of integration. Essentially, probability is included in the MATH 105 curriculum as an application of integration.