Section 9.1 Probability
Let \(S\) be a set of possible outcomes, called a sample space. Let \(E\) be a subset of \(S\) with some property. We call \(E\) an event.
The probability of event \(E\text{,}\) \(P(E)\text{,}\) is the number of outcomes in \(E\) divided by the total number of outcomes in \(S\text{.}\)
Let \(N(E)\) be the number of elements in \(E\text{.}\) Similarly, \(N(S)\text{,}\) is the number of outcomes in \(S\text{.}\)
In notation, \(P(E)=\frac{N(E)}{N(S)}\text{.}\)
A standard deck of playing cards has 52 cards. Each card has a suit and a value. The deck has four suits: hearts, diamonds, spades, clubs. Each suit has 13 values: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King. The spades and clubs are called “black” cards, the hearts and diamonds are called “red” cards.
Find the probability you randomly choose a club from the deck.
Note, \(E\) is the set of clubs, \(S\) is the set of cards.
Answer 1.\(P(E)=\frac{13}{52}=\frac{1}{4}\)
Find the probability you randomly choose a red card from the deck.
Note, \(E\) is the set of hearts and diamonds, \(S\) is the set of cards.
Answer 2.\(P(E)=\frac{26}{52}=\frac{1}{2}\)
Find the probability you randomly choose a 2 from the deck.
Note, \(E\) is the set of 2's, \(S\) is the set of cards.
Answer 3.\(P(E)=\frac{4}{52}=\frac{1}{13}\)
Activity 9.1.1.
Suppose you toss a fair coin two times.
(a)
List all the possible outcomes.
(b)
What is the probability that you get exactly 2 heads?
(c)
What is the probability that you get exactly 1 head?
In order to calculate a probability, we need to be able to count equally-likely events. Just because there are, say, 3 outcomes, that doesn't mean all the outcomes are equally-likely. For example, the Linfied football team could win, lose, or tie a game. Given the team's winning streak, the three outcomes are not equally-likely. They have a much higher probability of winning than just 1/3.
Thus, the focus for the rest of the chapter will be on counting events. Once we can count outcomes, it is straight-forward to find the probability.
Before moving on, though, we will state a few useful facts about probabilities.
- \(P(E)\) is always a number between 0 and 1. The probability can be 0.5. It cannot be 50%
- \(P(E)=0\) means the event is not possible.
- \(P(E)=1\) means the event always happens.
Reading Questions Check Your Understanding
1.
In a standard deck of cards, find the probability of randomly drawing an Ace. Give your answer to at least 3 decimal places.
The probability is .
2.
In a standard deck of cards, find the probability of randomly drawing a face card (Jack, Queen, King). Give your answer to at least 3 decimal places.
The probability is .
3.
In a standard deck of cards, find the probability of randomly drawing an even numbered card. Give your answer to at least 3 decimal places.
The probability is .
4.
An urn contains 3 red balls, 2 green balls, 4 multicolored balls. Find the probability of drawing a green ball. Give your answer to at least 3 decimal places.
The probability is .
5.
An urn contains 3 red balls, 2 green balls, 4 multicolored balls. Find the probability of drawing a solid colored ball. Give your answer to at least 3 decimal places.
The probability is .
Exercises Exercises
1.
Assume the sample space is a standard deck of 52 cards. Suppose you choose a random card from the deck.
- List all the possible outcomes of the card being red and not a face card.
- Calculate the probability of the event in (a).
2.
Assume the sample space is a standard deck of 52 cards. Suppose you choose a random card from the deck.
- List all the possible outcomes of the card being black and having an even number on it.
- Calculate the probability of the event in (a).
3.
Assume the sample space is the possible rolls of a pair of dice, one blue die and one red die.
- List all the possible outcomes where the sum of the numbers showing face up is 8.
- Calculate the probability of the event in (a).
4.
Assume the sample space is the possible rolls of a pair of dice, one blue die and one red die.
- List all the possible outcomes where the sum of the numbers showing face up is at most 6.
- Calculate the probability of the event in (a).
5.
Suppose a coin is tossed four times and the side showing face up on each toss is noted. Suppose also that heads and tails are equally-likely for the coin.
- List the 16 elements in the sample space as sequences of heads and tails.
- List the set of outcomes and the probability for the event of exactly one toss resulting in a head.
- List the set of outcomes and the probability for the event that at least two tosses result in a head.
- List the set of outcomes and the probability for the event that no toss results in a head.
6.
Suppose that on a true/false exam you have no idea about the answers to three questions. You choose your answers randomly and therefore have a 50-50 chance of being correct on any one question. For example, let \(CCW\) indicate that you were correct on the first two questions but wrong on the third.
- List all the elements of the sample space as the possible sequences of \(C\) and \(W\) for the three questions.
- List the set of outcomes and the probability for the event that exactly one answer is correct.
- List the set of outcomes and the probability for the event that at least two answers are correct.
- List the set of outcomes and the probability for the event that no answers are correct.
