# Definition of Probability
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## Section 1.2 Definition of Probability

### Definition 1.2.1.

A probability distribution on a sample space \(\Omega\) assigns probabilities to every event, satisfying the following conditions:

1. \(\Pr(\Omega) = 1\text{.}\)
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2. \(0 \leq \Pr(A) \leq 1\) for any event \(A\text{.}\)
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3. If \(A \cap B = \emptyset\text{,}\) then \(\Pr(A\cup B) = \Pr(A) + \Pr(B)\text{.}\)
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For small probability spaces (i.e., with finitely many outcomes in the sample space), we’ll usually assign probabilities to each individual outcome, and perhaps list them in a table. Then, to find the probability of any event, simply add together the probabilities of each outcome in that event.
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### Example 1.2.2.

An experiment consists of rolling a standard 6-sided die. The sample space is \(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\) The probability distribution (assuming a fair die) is shown below.
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Table 1.2.3. Distribution for a fair die
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\(x\) | \(\Pr(x)\)
1 | \(1/6\)
2 | \(1/6\)
3 | \(1/6\)
4 | \(1/6\)
5 | \(1/6\)
6 | \(1/6\)

One possible event is \(A = \{2, 4, 6\}\text{,}\) i.e., the event that the result of the roll is even. The probability of \(A\) is:
\begin{gather*} \Pr(A) = \Pr(2) + \Pr(4) + \Pr(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \end{gather*}
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### Definition 1.2.4.

Let \(\Omega = \{x_1, x_2, \dotsc, x_n\}\) be a sample space and \(A\subset \Omega\) an event. We refer to the distribution in which \(\Pr(x_i) = \frac{1}{n}\) for all \(i\) as the uniform distribution. In this case, it follows that \(\Pr(A) = \frac{|A|}{|\Omega|}\text{.}\)
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### Example 1.2.5.

A person picks a random number from 1 to 10. What is the probability that they picked 3?
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### Exercises Exercises

#### 1.

Consider the sample space \(\Omega = \{1, 2, 3, 4, 5, 6, 7, 8\}\) with probability distribution below. Calculate the probabilities of \(A = \{1, 3, 7, 8\}\text{,}\) \(B = \{2, 3, 6, 7\}\text{,}\) \(A\cup B\text{,}\) and \(A \cap B\text{.}\)
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Table 1.2.6.
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\(x\) | \(\Pr(x)\)
1 | 0.1
2 | 0.05
3 | 0.2
4 | 0.15
5 | 0.15
6 | 0.1
7 | 0.05
8 | 0.1
9 | 0.1

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#### 2.

Suppose we flip a coin two times. Answer the questions below. What about three flips? What about four flips?
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##### (a)

Write all outcomes in the sample space \(\Omega\text{.}\)
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##### (b)

Make a probability distribution table for \(\Omega\) assuming the coin is fair.
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##### (c)

Make a probability distribution table assuming the coin comes up heads with probability 0.3.
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#### 3.

Suppose we roll a die two times. Answer the questions below.
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##### (a)

Write all outcomes in the sample space \(\Omega\text{.}\)
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##### (b)

Make a probability distribution table for \(\Omega\) assuming the die is fair.
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##### (c)

Let \(A\) be the event that the second roll is higher than the first, and let \(B\) be the event that the first roll is even. Find \(\Pr(A), \Pr(B)\text{,}\) and \(\Pr(A \mid B)\text{.}\)
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#### 4.

Suppose a die has the values \(1, 2, 3, 4, 5, 6\) on the faces, but the die is not fair. Instead, the probabilities scale by the same amount as the face values. For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on. Write a probability distribution table for this die.
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#### 5.

Suppose a die has the values \(1, 2, 3, 4, 5, 6\) on the faces, but the die is not fair. Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll. Write a probability distribution table for this die.
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#### 6.

A toxin molecule inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. For each value of \(n = 1, 2, 3, \dotsc\text{,}\) find the probability of the toxin molecule leaving the cell during the \(n\)th minute. What is the probability of the molecule leaving the cell during the first 3 minutes?
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#### 7.

Each of 10 toxin molecules inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. For each value of \(n = 1, 2, 3, \dotsc\text{,}\) and for each value of \(0\leq k \leq n\text{,}\) find the probability that exactly \(k\) toxin molecules remain in the cell after the \(n\)th minute.
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