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Section 11.2 – Probability
*The probability, p, of an event is a number such that 0 ≤ p ≤ 1
The probability of an impossible event is 0 (0 %), while the probability of a certain event is 1 (100%)
Experimental Probability of event: P(event) = number of times the event occursnumber of trials
Example 1 – Of the 60 vehicles in a certain parking lot, 15 are pickup trucks. What is the experimental probability that a vehicle in the lot is a pickup truck?
Example 2 - A baseball player got a hit in 20 of his last 55 at bats. What is the experimental probability that he will get a hit in his next at bat?
*Using a simulation to predict
(Page 682, Problem #2) – On a multiple-choice test, each item has 4 choices, but only one choice is correct. How can you simulate guessing the answers? What is the probability that you will pass the test by guessing at least 6 or 10 answers correctly?
*The set of all possible outcomes to an experiment or activity is the sample space. When these outcomes have the same chance of occurring, the outcomes are known as equally likely outcomes.
Example – A six-sided dice has 6 equally likely outcomes. The chance of a 1, 2, 3…, and 6 is just as likely to be rolled as any other number.
Theoretical Probability – If a sample space has n equally likely outcomes and an event A occurs in m of these outcomes, then the theoretical probability of event A is P(A) = mn
Example – What is the theoretical probability of getting a 5 on one roll of a standard 6-sided dice?
There are six equally likely outcomes, 1, 2, 3, 4, 5, and 6. A 5 occurs one way.
P(5) = 16
Problem 1 - What is the theoretical probability of getting a sum of 5 on one roll of two 6-sided dice?
Problem 2 – What is the probability of getting a sum that is an odd number on one roll of two 6-sided dice?
Problem 3 – A jar contains 30 red marbles, 50 blue marbles and 20 white marbles. You pick one marble from the jar at random. Find each theoretical probability.
P(red) =
P(blue) =
P(not white) =
P(red or blue) =
Bellringer: What is the theoretical probability of rolling two 6-sided dice and the sum of the dice faces being equal to or greater than 8.
*Sometimes it can be easier to use combinatorics to find theoretical probability rather than counting all of the possible outcomes (like when working with dice in previous examples).
Example – What is the theoretical probability of being dealt exactly two 7’s in a 5-card hand from a standard 52-card deck?
Combinations of two 7’s from four 7’s: 4C2
Combinations of three non-sevens from other 48 cards: 48C3
Number of 5-card hands with two 7’s: 4C2 · 48C3
Number of possible 5-card hands from 52 cards: 52C5
P(hand with two 7’s): 4C2 · 48C352C5 = 103,7762,598,960 ≈ 0.0399 or about 4%
Example 1 – What is the theoretical probability of being dealt all four 7’s in a 5-card hand?
Example 2 – What is the theoretical probability of being dealt exactly three Kings, one Ace, another random card from a standard 52-card deck?
Geometric Probability
Example - A certain batter has a strike zone of 22” x 17”. His high-inside strike zone is a 6” x 4” area. What is the chance that a random strike is in his high-strike zone area?
P(high-inside strike) = area of high-inside strike zonearea of total strike zone = 4 ·617 ·22 ≈ 0.064 or about 6.4%
Example 2 - What if the low-outside strike zone is also 6” x 4”?
What is the difference between experimental probability and theoretical probability?
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Why is simulation better the more times you perform it?
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