Applicable Mathematics/Probability Distributions

Probability Distribution

Some experiments have numerical outcomes, like rolling a die for instance. A variable whose value is the numerical outcome of a random event is called a random variable. Take rolling a die, for example. We can let the random variable D represent the number showing on the die when rolling the die. Then, D equals either 1, 2, 3, 4, 5, or 6.

A function that puts together a probability with its outcome in an experiment is known as a probability distribution. Or, another way of putting it is that it is a function that maps the sample space to the probabilities of the outcomes in the sample space for a particular random variable. The numbers below illustrate the probability distribution for rolling a die.

D = roll 1 2 3 4 5 6

Probability 1/6 1/6 1/6 1/6 1/6 1/6

P(D = 4) = 1/6

A uniform distribution is a distribution where all of the probabilities are the same. The probability distribution above has uniform distribution.

The use of a table of probabilities (or a graph) can help you visualize a probability distribution. Such graphs are known as relative- frequency histograms.

Example of Probability Distribution

Suppose 2 dice are rolled. The table shows the distribution of the sum of the numbers rolled.

S = Sum 2 3 4 5 6 7 8 9 10 11 12 Probability 1/36 1/18 1/12 1/9 5/36 1/6 5/36 1/9 1/12 1/18 1/36

a. Use the table to find P(S = 10). What other sum has the same probability?

The probability of the sum of 9 (according to the table) is 1/12. The other sum of 4 has the same probability of 1/12.

b. What are the odds of rolling a sum of 8?

Step 1 Identify s and f.

```             P(rolling an 8) = 5/36

= s/(s + f)     s = 5, 36－5=31， f = 31
```

Step 2 Find the odds.

```              Odds = s:f
```
```                   = 5:31
```

So, the odds of rolling a sum of 8 are 5:31.