11-2 Basic Probability

Theoretical vs. Experimental Theoretical probabilities are those that can be determined purely on formal or logical grounds, independent of prior experience. Experimental probabilities are estimates of the relative frequency of an event based by our past observational experience.

Experimental Probability

Example 1—Experimental Probability Out of the 60 vehicles in the teacher parking lot today, 15 are pickup trucks. What is the probability that a vehicle in the lot is a pickup truck? GOT IT? A softball player got a hit in 20 of her last 50 times at bat. What is the probability that she will get a hit in her next at bat?

Probability Experiment Can be repeated many times (at least in theory) Each such repetition is called a trial When an experiment is performed it can result in one or more outcomes, which are called events.

Law of Large Numbers The more repetitions we take, the more closely the experimental probability will reflect the true theoretical probability. This is sometimes referred to as the Law of Large Numbers, which states that if an experiment is repeated a large number of times, the relative frequency of the outcome will tend to be close to the theoretical probability of the outcome.

Summary of 20,000 Coin Tosses Num of Tosses Num. of Heads Relative Freq. n f f / n 10 8 .8000 100 62 .6200 1,000 473 .4730 5,000 2,550 .5100 10,000 5,098 .5098 15,000 7,649 .5099 20,000 10,038 .5019

Example 2—A Simulation On a multiple choice test, each item has four choices, but only one choice is correct. What is the probability that we will pass the test by guessing? (6 out of 10 OR 60%) How can we simulate guessing? How many trials should we run?

Choose the numbers from 1-4, let 1 represent the correct solution and use Using a Random # generator. P(passing) = 1/16

Theoretical Probability

Example 3—Finding Theoretical Probability What is the probability of getting a 3 on one roll of a standard number cube? What is the probability of getting a sum of 4 on one roll of two standard number cubes? What is the probability of getting a sum that is an odd number on one roll of two standard number cubes? P(3) = 1/6 P(sum 4) = 3/36= 1/12 P(sum is odd) = 1/2

Geometric Probability

Solution—Use Area

Your Turn—Geometric Probability A carnival game consists of throwing darts at a circular board as shown. What is the geometric probability that a dart thrown at random will hit the shaded circle? ANSWER: About 14%

11-3 Probability of Multiple Events

Outcomes of Different Events When the outcome of one event affects the outcome of a second event, we say that the events are dependent. When one outcome of one event does not affect a second event, we say that the events are independent.

Decide if the following are dependent or independent An expo marker is picked at random from a box and then replaced. A second marker is then grabbed at random. Two dice are rolled at the same time. An Ace is picked from a deck of cards. Without replacing it, a Jack is picked from the deck. Independent Independent Dependent

How to find the Probability of an “AND” for Two Independent Events Ex: If P(A) = ½ and P(B) = 1/3 then P(A and B) = Ex: If P(A) = ½ and P(B) = 1/3 then P(A and B) =

Your Turn A box contains 20 red marbles and 30 blue marbles. A second box contains 10 white marbles and 47 black marbles. If you choose one marble from each box without looking, what is the probability that you get a blue marble and a black marble? The probability that a blue and a black marble will be drawn is , or 49%. 47 95

Probability of an A “OR” B Independent Events Exclusive Events If Two events are mutually exclusive then they can not happen at the same time.

Hint: Is it impossible for the events to occur at the same time? Mutually Exclusive?? A spinner has ten equal-sized sections labeled 1 to 10. Find the probability of each event. P(even or multiple 0f 5) B. P(Multiple of 3 or 4) No! P(A)+P(B)-P(A and B) Yes! P(A)+P(B)

Work them through… Do you know what to do?? A spinner has ten equal-sized sections labeled 1 to 10. Find the probability of each event. P(even or multiple 0f 5) B. P(Multiple of 3 or 4)

Probability of Multiple Events A spinner has twenty equal-size sections numbered from 1 to 20. If you spin the spinner, what is the probability that the number you spin will be a multiple of 2 or a multiple of 3? No. So: P(A) + P(B) - P(A and B) Are the events mutually exclusive?

FSA Practice Problems of the Day!