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. You can convert proportions to percentages by multiplying by 100. For example, .1 = 10%.

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).

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).

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

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).

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).

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

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.

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).

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.

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)

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

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.

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).

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

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).

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.

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

Homework Study for Chapter 7 Quiz Read Chapter 8 Do Chapter 7 HW