Probability The risk of getting struck by lightning in any year is 1 in 750,000. The chances of surviving a lightning strike are 3 in 4. These risks and chances are a way of describing the probability of an event. The probability of an event is a ratio that measures the chances of the event occur.

Introduction to Probability The Probability of an event occurring is a number between 0 and 1. 0 means that the event cannot occur… 1 means that the event is certain to occur. The probability that event A will occur is P(A) = number of outcomes in A total number of outcomes

P(3)= P(prime #)= P(factors of 8)= P(9)= 1 2 3 4 5 6 7 8 Quick Examples of Probability: 1 2 3 4 5 6 7 8 P(3)= P(prime #)= P(factors of 8)= P(9)=

Ex1: a. Four cards are drawn from a standard 52-card deck. What is the probability that the first three cards are black? b. Five cards are drawn from a standard 52-card deck. What is the probability that the first five cards are spades?

Roman has a collection of 26 books–16 are fiction and 10 are nonfiction. He randomly chooses 8 books to take with him on vacation. What is the probability that he chooses 4 fiction and 4 nonfiction? Step 1 Determine how many 8-book selections meet the conditions. C(16, 4) Select 4 fiction books. Their order does not matter. C(10, 4) Select 4 nonfiction.

Answer: The probability is about 0.24464 or 24.5%. Example 3-2a

Another way to measure the chance of an event occurring is with odds. Odds: The odds that an event will occur can be expressed as the ratio of the number of successes to the number of failure. Odds of success = s : f

a. What are the odds that a male born in 1980 will live to age 65? Example 3 Life Expectancy The chances of a male born in 1980 to live to be at least 65 years of age are about 7 in 10. For females, the chances are about 21 in 25. a. What are the odds that a male born in 1980 will live to age 65? b. What are the odds that a female born in 1980 will live to age 65? Answer: 7:3 Answer: 21:4 Example 3-3b

Many experiments, such as rolling a die, have numerical outcomes. A random variable is a variable whose value if the numerical outcome of a random event. For example, when rolling a die we can let the random variable D represent the number showing on the die. Thus, D can equal 1, 2, 3, 4, 5, or 6. A probability distribution, for a particular random variable is a function that maps the sample space to the probabilities of the outcomes in the sample space.

The following is the probability distribution for rolling a die. D = Roll 1 2 3 4 5 6 Probability Further, to help visualize a probability distribution, you can a graph, called a relative-frequency histogram.

Relative-frequency histogram Suppose two dice are rolled. The table and the relative-frequency histogram show the distribution of the sum of the numbers rolled. Probability 12 11 10 9 8 7 6 5 4 3 2 S = Sum Example 3-4a

Relative-frequency histogram Use the graph to determine which outcomes are least likely. What are their probabilities? Answer: The least probability is The least likely outcomes are sums of 2 and 12. Example 3-4a

Use the table to find What other sum has the same probability. 12 11 10 9 8 7 6 5 4 3 2 S = Sum Use the table to find What other sum has the same probability. Answer: The other sum with this probability is a sum of 3. Example 3-4a

What are the odds of rolling a sum of 5? Step 1 Identify s and f. Step 2 Find the odds. Answer: So, the odds of rolling a sum of 5 are 1:8. Example 3-4a