# 6.1. Sample Spaces and Probability

**Sample Spaces and Probability**

If two coins are tossed, what is the probability that both coins will fall heads? The problem seems simple enough, but it is not uncommon to hear the incorrect answer of . A student may incorrectly reason that if two coins are tossed there are three possibilities, one head, two heads, or no heads. Therefore, the probability of two heads is one out of three. The answer is wrong because if we toss two coins there are four possibilities and not three. For clarity, assume that one coin is a penny and the other a nickel. Then we have the following four possibilities:

HH HT TH TT

The possibility HT, for example, indicates a head on the penny and a tail on the nickel, while TH represents a tail on the penny and a head on the nickel.

It is for this reason, we emphasize the need for understanding sample spaces.

An act of flipping coins, rolling dice, drawing cards, or surveying people are referred to as an **experiment**.

**Sample Spaces**: A sample space of an experiment is the set of all possible outcomes.

Example 6.1.1

**Solution**

**S**is:

Example 6.1.2

**Solution**

We illustrate these possibilities with a tree diagram:

Example 6.1.3

**Solution**

We assume one of the dice is red, and the other green. We have the following 36 possibilities:

Green | ||||||

Red | 1 | 2 | 3 | 4 | 5 | 6 |

1 | (1, 1) | (1, 2) | (1, 3) | (1, 4) | (1, 5) | (1, 6) |

2 | (2, 1) | (2, 2) | (2, 3) | (2, 4) | (2, 5) | (2, 6) |

3 | (3, 1) | (3, 2) | (3, 3) | (3, 4) | (3, 5) | (3, 6) |

4 | (4, 1) | (4, 2) | (4, 3) | (4, 4) | (4, 5) | (4, 6) |

5 | (5, 1) | (5, 2) | (5, 3) | (5, 4) | (5, 5) | (5, 6) |

6 | (6, 1) | (6, 2) | (6, 3) | (6, 4) | (6, 5) | (6, 6) |

Now that we understand the concept of a sample space, we will define probability.

**Probability**: For a sample space *S*, and an outcome *A* of *S*, the following two properties are satisfied.

- If
*A*is an outcome of a sample space, then the probability of*A*, denoted by*P*(*A*), is between 0 and 1, inclusive. 0 ≤*P*(*A*) ≤ 1 - The sum of the probabilities of all the outcomes in
*S*equals 1.

Example 6.1.4

**Solution**

The example we just considered consisted of only one outcome of the sample space. We are often interested in finding probabilities of several outcomes represented by an event.

An **event** is a subset of a sample space. If an event consists of only one outcome, it is called a **simple event**.

Example 6.1.5

**Solution**

*E*represent the event that the sum of the faces of two dice is 7. Since the possible cases for the sum to be 7 are: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1).

*E*= {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1)}

*P*(

*E*) = 6/36 or 1/6.

Example 6.1.6

**Solution**

*r*

_{1},

*r*

_{2},

*r*

_{3},

*w*

_{1},

*w*

_{2},

*w*

_{3},

*w*

_{4},

*b*

_{1},

*b*

_{2},

*b*

_{3}. Let the event

*C*represent that the marble is red or blue.

*S*= {

*r*

_{1},

*r*

_{2},

*r*

_{3},

*w*

_{1},

*w*

_{2},

*w*

_{3},

*w*

_{4},

*b*

_{1},

*b*

_{2},

*b*

_{3}}

*C*= {

*r*

_{1},

*r*

_{2},

*r*

_{3},

*b*

_{1},

*b*

_{2},

*b*

_{3}}

*C:*

*P*(

*C*) = 6/10 or 3/5.

Example 6.1.7

**Solution**

*F*represent that the sum of the numbers is four. Then:

*P*(

*F*) = 2/6 or 1/3.

Example 6.1.8

*at least*4?

**Solution**

*S*= {(1, 2), (1, 3), (2, 3), (2, 1), (3, 1), (3, 2)}

*A*represent that the sum of the numbers is at least four. Then:

*F*= {(1, 3), (3, 1), (2, 3), (3, 2)}

*F*is:

*P*(

*F*) = 4/6 or 2/3.

**Practice questions**

**1.** Write a sample space for the following event: a die is rolled, and a coin is tossed.

**2. **A card is selected from a deck of 52 playing cards. Find the following probabilities:

**a**. *P* (a king)

**b.** *P* (any suit other than hearts)

**3. **A jar contains 6 red, 7 white, and 7 blue marbles. If a marble is chosen at random, find the following probabilities:

**a**. *P* (red)

**b.** *P* (red or blue)

**4. **Two dice are rolled. Find the following probabilities:

**a**. *P* (the sum of the dice is 5)

**b.** *P* (the sum of the dice is 3 or 6)

**5. **A family has four children. Find the following probabilities:

**a**. *P* (they have two boys and two girls)

**b.** *P* (they have three or more girls)