Probability, Discrete Probability, Normal Probabilty
- 2. Contents Definition of Probability The Probability Distribution a) Discrete Probability Distributions b) Normal Probability Distributions
- 3. Definition of Probability • The term “probability” in Statistics refers to the chances of occurrence of an event among a large number of possibilities. OR • When all the equally possible occurrences have been enumerated, the probability Pr of an event happening is the ratio of the number of ways in which the particular event may occur to the total number of possible occurrences.
- 4. The Probability Distribution • Discrete Probability Distributions • Normal Probability Distributions
- 5. Discrete Probability Distributions • Random Variables A random variable X represents a numerical value associated with each outcome of a probability distribution. A random variable is discrete if it has a finite or countable number of possible outcome that can be listed. A random variable is continuous if it has an uncountable number or possible outcome, represented by the intervals on a number line.
- 6. Random Variables Example: Decide if the random variable X is discrete or continuous. • The distance your car travels on a tank of gas The distance your car travels is a continuous random variable because it is a measurement that cannot be counted. (All measurements are continuous random variables.) • The number of students in a statistics class The number of students is a discrete random variable because it can be counted.
- 7. Discrete Probability Distributions • A discrete probability distribution lists each possible value the random variable can assume, together with its probability. A probability distribution must satisfy the following conditions. In Words In Symbols a. The probability of each value of the discrete random variable is between 0 and 1, inclusive. 0 P (x) 1 b. The sum of all the probabilities is 1. ΣP (x) = 1
- 8. Constructing a Discrete Probability Distribution Guidelines Let x be a discrete random variable with possible outcomes x1, x2, …, xn. • Make a frequency distribution for the possible outcomes. • Find the sum of the frequencies. • Find the probability of each possible outcome by dividing its frequency by the sum of the frequencies. • Check that each probability is between 0 and 1 and that the sum is 1.
- 9. Constructing a Discrete Probability Distribution Example: The spinner below is divided into two sections. The probability of landing on the 1 is 0.25. The probability of landing on the 2 is 0.75. Let X be the number the spinner lands on. Construct a probability distribution for the random variable X.
- 10. Constructing a Discrete Probability Distribution Example: The spinner below is Spun two times. The probability of landing on the 1 is 0.25. The probability of landing on the 2 is 0.75. Let X be the sum of the two spins. Construct a probability distribution for the random variable X.
- 11. Constructing a Discrete Probability Distribution Example Continued: 2
- 12. Constructing a Discrete Probability Distribution Example Continued: 2
- 14. Properties of Normal Distributions • A continuous random variable has an infinite number of possible values that can be represented by an interval on the number line. The time spent studying can be any number between 0 and 24. • The probability distribution of a continuous random variable is called a continuous probability distribution. Hours spent studying in a day 0 63 9 1512 18 2421
- 15. Properties of Normal Distributions • The most important probability distribution in statistics is the normal distribution. x Normal curve • A normal distribution is a continuous probability distribution for a random variable, x. The graph of a normal distribution is called the normal curve.
- 16. Properties of Normal Distributions i. The mean, median, and mode are equal. ii. The normal curve is bell-shaped and symmetric about the mean. iii. The total area under the curve is equal to one. iv. The normal curve approaches, but never touches the x-axis as it extends farther and farther away from the mean. v. Between μ σ and μ + σ (in the center of the curve), the graph curves downward. The graph curves upward to the left of μ σ and to the right of μ + σ. The points at which the curve changes from curving upward to curving downward are called the inflection points.
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