April 20, 2015
Chapter 6 - Probability
Section 6.1 The Idea of Probability
Section 6.2 Probability Models
Section 6.3 General Probability Rules
April 20, 2015
Simple Question If tossing a coin, what is the probability of it turning up heads?
How did you come up with your answer?
April 20, 2015
Probability
• Probability is the branch of mathematics that describes the pattern
of chance outcomes.
• The heart of statistics is statistical inference.
(Statistical Inference: a method by which to answer specific
questions with a known degree of confidence.)
Probability calculations are the basis for inference.
• Mathematical probability is an idealization based on imagining what
would happen in an indefinitely long series of trials.
April 20, 2015
Random Behavior
• Many observable phenomena are random: the relative frequencies of
outcomes seem to settle down over the long haul.
• The big idea of probability: chance ("random") behavior is unpredictable
in the short run, but has regular and predictable patterns in the long run.
April 20, 2015
Example 6.1, pg 331
When you toss a coin, there are only two possible outcomes, heads or tails. Figure 6.1 shows
the results of tossing a coin 1000 times. For each number of tosses from 1 to 1000, we have
plotted the proportion of those tosses that came up heads. The first toss was a head, so the
proportion of heads starts at 1. The second toss was a tail, reducing the proportion of heads to
0.5 after two tosses. The next three tosses gave a tail followed by two heads, so the proportion
of heads after five tosses is 3/5 or 0.6. The proportion of tosses that produce heads is quite
variable at first, but it settles down as we make more and more tosses. Eventually this
proportion gets close to 0.5 and stays there. We say that 0.5 is the probability of a head. The
probablity 0.5 appears as a horizontal line on the graph.
Figure 6.1, pg 331
April 20, 2015
Simulation Problem - Exercise 6.3, pg 334
The basketball player Shaquille O'Neal makes about half of
his free throws over an entire season. We will use teh
calculator to simulate 100 free throws shot independently by a
player who has probability 0.5 of making each shot. We let the
number 1 represent the outcome "Hit" and 0 represent the
outcome "Miss."
(a) Enter the command randInt(0, 1, 100)
L1. This tells the
calculator to randomly select a hit (1) or a miss (0), do this
100 times in succession, and store the results in the list L1.
(b) What percent of the 100 shots are hits?
(c) Examine the sequence of hits and misses. How long was
the longest run of shots made? Of shots missed? Sequences
of random outcomes often show runs longer than our intuition
thinks likely.
April 20, 2015
Probability Definitions
Sample Space (S) - the set of all possible outcomes
see Example 6.3, pg 336 (rolling two dice)
Event - a particular outcome or set of outcomes; a subset of the sample
space.
Probability - the number of times an event can occur within the sample space
divided by the sample space.
April 20, 2015
Figure 6.2, pg 336
The 36 possible outcomes of rolling two dice
What is the probability of getting a sum of ≤ 5 when rolling two dice?
April 20, 2015
Sample Space (S)
Three ways to represent a sample space:
• list all the possible outcomes (as in the two dice example)
• create a tree diagram (figure 6.3, pg 338)
• create a Venn diagram (figure 6.4, pg 343)
Ways to calculate the number of outcomes in the Sample Space
Multiplication Principle - If you can do one task in "a" number of ways, and a
second task in "b" number of ways, then both tasks can be done in a x b
number of ways.
Tree Diagram - to illustrate all the possible outcomes
Example 6.5 - Flip a Coin and Roll a Die
An experiment consists of flipping a coin and
rolling a die. Posible outcomes are head (H)
followed by any of the digits 1 to 6, or a tail (T)
followed by any of the digits 1 to 6. The sample
space consists of 12 outcomes. Illustrated in
figure 6.3.
figure 6.3, pg 338
April 20, 2015
Homework due Friday 4/24
Read pgs 335 - 352 and
Complete Exercises:
6.11 pg. 340
6.14 parts (a) and (b), pg 341
6.18, pg 342