1 2 4 3
Experiment An experiment is an occurrence, leading to one of several specified outcomes, whose result cannot be known ahead of time.
Sample Space A sample space is the set of all possible outcomes of an experiment.
Trial A trial is a single observation.
Outcome An outcome is the result of a single trial.
Event An event is a possible outcome of a trial.
number of favorable outcomes number of possible outcomes Probability Formula The probability of an event E is P(E) = . number of favorable outcomes number of possible outcomes
Example 1 For one spin, find each of the following probabilities (the number in parentheses is the number on which the spinner stops). Express the probability ratio as both a fraction in lowest terms and a decimal.
favorable outcomes possible outcomes 3 2 1 Find P(3). favorable outcomes possible outcomes 1 6 = ≈ 0.17
favorable outcomes possible outcomes 3 2 1 Find P(2). favorable outcomes possible outcomes 3 6 = = 0.5
favorable outcomes possible outcomes 3 2 1 Find P(1). favorable outcomes possible outcomes 2 6 = ≈ 0.33
Example 2 For one spin, find each of the following probabilities (x is the number on which the spinner stops).
favorable outcomes possible outcomes 2 1 3 Find P(x < 4). 6 4 5 favorable outcomes possible outcomes 3 6 = = 0.5
favorable outcomes possible outcomes 2 1 3 Find P(x < 7). 6 4 5 favorable outcomes possible outcomes 6 6 = = 1
favorable outcomes possible outcomes 2 1 3 Find P(x ≥ 3). 6 4 5 favorable outcomes possible outcomes 4 6 = ≈ 0.67
Example Twelve large blue and thirteen large red marbles are placed in a bag with fifteen small blue, seven small red, and three small white marbles. Find each of the following probabilities.
P(large) = 0.5 P(red) = 0.4 P(blue) = 0.54
Probability of Mutually Exclusive Events If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B).
Example 3 For one spin, find each of the following probabilities (x is the number on which the spinner stops, and the color denotes the color of the wedge on which the spinner stops).
Find P(white or x > 4). 5 1 4 3 2
Find P(white or x > 4). E1 = spinner stopping on a white wedge E2 = spinner stopping on a number greater than 4 They are mutually exclusive.
Find P(white or x > 4). 2 5 = 1 5 = P(white) P(x > 4) = 0.4 = 0.2 P(white or x > 4) = P(white) + P(x > 4) 2 5 = 1 5 + 3 5 = = 0.6
Find P(white or even). 5 1 4 3 2
Find P(white or even). E1 = spinner stopping on a white wedge E2 = spinner stopping on an even number They are mutually exclusive.
Find P(white or even). 2 5 = 2 5 = P(white) P(even) = 0.4 = 0.4 P(white or even) = P(white) + P(even) 2 5 = + 4 5 = = 0.8
Example 4 If one of the number squares pictured is drawn at random, find each of the following probabilities.
Find P(white or blue). 2 4 6 8 1 3 5 7 10 20 30 40
Find P(white or blue). Drawing a white square and drawing a blue square are mutually exclusive events. P(white or blue) = P(white) + P(blue) 4 12 = + 8 12 = 2 3 = ≈ 0.67
Find P(multiple of 5 or white). 2 4 6 8 1 3 5 7 10 20 30 40
Find P(multiple of 5 or white). Drawing a multiple of 5 and drawing a white square are mutually exclusive events. P(multiple of 5 or white) = P(multiple of 5) + P(white) 5 12 = 4 12 + 9 12 = 3 4 = = 0.75
Example Twelve large blue and thirteen large red marbles are placed in a bag with fifteen small blue, seven small red, and three small white marbles. Find each of the following probabilities.
P(large or blue) = 0.8 P(small or red) = 0.76 P(small or large) = 1
P(large and blue) = 0.24 P(small and red) = 0.14 P(large and white) = 0
Exercise Use the definition of probability, the Fundamental Principle of Counting, and the idea of combinations to answer the following problems. Leave answers in symbol form.
If a committee of five students is selected from a class of thirteen boys and fifteen girls, what is the probability that the committee will have three girls on it? 15C3 × 13C2 28C5
What is the probability that the committee will have no boys on it? 15C5 28C5
What is the probability that the committee will have four boys on it? 15C1 × 13C4 28C5