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Probability

Probability. Definition. Probability: the proportion of times an event would occur if the chances for occurrence were infinite ( p . 123) A proportion is a number from 0 to 1. In probability, 1 means that the event will certainly occur, 0 means that the event will certainly not occur.

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Probability

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  1. Probability

  2. Definition • Probability: the proportion of times an event would occur if the chances for occurrence were infinite (p. 123) • A proportion is a number from 0 to 1. • In probability, 1 means that the event will certainly occur, 0 means that the event will certainly not occur. • You can convert proportions to percentages by multiplying by 100. For example, .1 = 10%.

  3. Probability • Symbolized by p. • If you are rolling a die, the probability of rolling a two can be expressed as p(2). • The probability of an event’s occurrence plus the probability of its nonoccurrence must add up to 1 (it’s either going to happen or it isn’t).

  4. The Addition Rule • The probability of one event occurring OR another event occurring can be determined by adding the two (or more) probabilities. • The probability of flipping a head OR a tail is: • p(H) = .5 • p(T) = .5 • p(H or T) = p(H) + p(T) = .5 + .5 = 1 • So, it is certain (ignoring other possibilities).

  5. The Addition Rule The Addition Rule of Probability The “OR” Rule p(A or B) = p(A) + p(B)

  6. You Try • What is the probability of drawing a red bean OR a black bean from my cup ‘o’ beans? • There are 50 red beans, 50 black beans, and 50 white beans. • Write your solution on a piece of paper, and hang on to it (show your work).

  7. The Multiplication Rule • The probability of two or more independent events occurring on separate occasions can be determined by multiplying the individual probabilities. • The probability of flipping a head and then flipping tail is: • p(H) = .5 • p(T) = .5 • p(H, T) = p(H) × p(T) = .5 × .5 = .25 • So there is a 25% chance of flipping a head then a tail (in theory).

  8. The Multiplication Rule The Multiplication Rule of Probability The “AND” Rule p(A, B) = p(A) × p(B)

  9. You Try • What is the probability of drawing a red bean and then a black bean (after putting the red bean back)? • There are 50 red beans, 50 black beans, and 50 white beans. • Write your solution on the same paper.

  10. Independent Events • We have been talking about independent events, events that do not influence the probability of another event. • Flipping a coin and sampling with replacement are examples of independent events (the probability remains the same).

  11. Conditional Probability • Definition: The probability of an event, given that another event has already occurred. • It is similar to the “AND” rule except that we have to account for the occurrence of the preceding event.

  12. Conditional Probability • What is the probability of drawing the queen of hearts and then drawing the queen of spades (without putting the queen of hearts back)? • p(QH, QS) = p(QH) × p(QS|QH) • Read p(QS|QH) as “the probability of queen of spades given queen of hearts.” • p(QH, QS) = 1/52 × 1/51 = 1/2,652 (.0004)

  13. Conditional Probability For more than two events… p(A, B, C) = p(A) × p(B|A) × p(C|A, B)

  14. You Try • What is the probability of drawing a black bean, then a red bean, and then a white bean (without putting any of them back)? • There are 50 red beans, 50 black beans, and 50 white beans. • What is the probability of drawing a black bean, then a red bean, and then a blue bean? • Write your answers on your paper.

  15. Conditional Probability • Using basic algebra, we can also modify our conditional probability formula to solve for p(B|A). • Instead of p(A, B) = p(A) × p(B|A) • We can write p(B|A) = p(A, B)/p(A) • We divided both sides by p(A).

  16. Conditional Probability • For example, if you know that the probability of a randomly selected individual being a male who smokes p(M, S) is .19, and you know that the probability of randomly selecting a male p(M) is .6, you can determine the probability of a randomly selected male smoking by using the formula: • P(S|M) = p(M, S)/p(M) • P(S|M) = .19/.6= .32

  17. You Try • If we know the probability of selecting a prisoner in the “high suicide risk” category who has attempted suicide is .1, and we know that the probability of selecting a prisoner in the “high suicide risk” category is .25, if we just randomly selected a prisoner in the “high suicide risk” category, what is the probability that they have attempted suicide? • Write your answer on your paper (show your work).

  18. Binomial Distribution • When you only have two possible outcomes (like a coin flip), certain outcome sequences are more probable than others. • For example, if I flip a coin three times, there is a .375 probability that I will flip heads one or two times, but only a .125 probability that I will flip heads three or zero times.

  19. Binomial Distribution • If we create a bar graph with the relative frequencies (probabilities) of each outcome, it will look like this:

  20. Homework • Study for Chapter 7 Quiz • Read Chapter 8 • Do Chapter 7 HW

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