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The Shape of Distributions Symmetric Distribution • A vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images.

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The Shape of Distributions Uniform Distribution (rectangular) • All entries or classes in the distribution have equal or approximately equal frequencies. • Symmetric.

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The Shape of Distributions Skewed Left Distribution (negatively skewed) • The “tail” of the graph elongates more to the left.

• The mean is to the left of the median.

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The Shape of Distributions Skewed Right Distribution (positively skewed) • The “tail” of the graph elongates more to the right.

• The mean is to the right of the median.

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Chapter 4 Probability

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Basic Concepts of Probability

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Types of Probability Classical (theoretical) Probability • Each outcome in a sample space is equally likely.

Number of outcomes in event E • P( E )  Number of outcomes in sample space

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Example: Finding Classical Probabilities You roll a six-sided die. Find the probability of each event.

1. Event A: rolling a 3 2. Event B: rolling a 7 3. Event C: rolling a number less than 5 Solution: Sample space: {1, 2, 3, 4, 5, 6}

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Solution: Finding Classical Probabilities 1. Event A: rolling a 3

Event A = {3}

1 P(rolling a 3)   0.167 6

2. Event B: rolling a 7 0 P(rolling a 7)   0 6

Event B= { } (7 is not in the sample space)

3. Event C: rolling a number less than 5 Event C = {1, 2, 3, 4} 4 P(rolling a number less than 5)   0.667 6 9

Types of Probability Empirical (statistical) Probability • Based on observations obtained from probability experiments.

• Relative frequency of an event.

Frequency of event E f  • P( E )  Total frequency n

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Example: Finding Empirical Probabilities A company is conducting an online survey of randomly selected individuals to determine if traffic congestion is a problem in their community. So far, 320 people have responded to the survey. What is the probability that the next person that responds to the survey says that traffic congestion is a serious problem in their community?

Response

Number of times, f

Serious problem

123

Moderate problem

115

Not a problem

82 ÎŁf = 320 11

Solution: Finding Empirical Probabilities Response

event

Number of times, f

Serious problem

123

Moderate problem

115

Not a problem

82

frequency

Σf = 320

f 123 P( Serious problem)    0.384 n 320

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Types of Probability Subjective Probability • Intuition, educated guesses, and estimates. • e.g. A doctor may feel a patient has a 90% chance of a full recovery.

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Example: Classifying Types of Probability Classify the statement as an example of classical, empirical, or subjective probability.

1. The probability that you will be married by age 30 is 0.50.

Solution: Subjective probability (most likely an educated guess)

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Example: Classifying Types of Probability Classify the statement as an example of classical, empirical, or subjective probability.

2. The probability that a voter chosen at random will vote Republican is 0.45.

Solution: Empirical probability (most likely based on a survey)

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Example: Classifying Types of Probability Classify the statement as an example of classical, empirical, or subjective probability.

3. The probability of winning a 1000-ticket raffle with 1 one ticket is 1000.

Solution: Classical probability (equally likely outcomes)

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‫هذا الجزء للطالب الذين يعانىن من‬ ‫مشاكل في قراءة الكتاب وصعىبة الفهم‬

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Conditional Probability and the Multiplication Rule

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Conditional Probability Conditional Probability • The probability of an event occurring, given that another event has already occurred • Denoted P(B | A) (read “probability of B, given A”)

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