Basic Probability CCM2 Unit 6: Probability

Basic Probability Probability of an event occurring is: P(E) = Number of Favorable Outcomes Total Number of Outcomes Your answer can be written as a fraction or a %. (Remember to write it as a %, you need to multiply the decimal by 100) We can use sample spaces, intersections, unions, and compliments of sets to help us find probabilities of events. Note that P(AC) is every outcome except (or not) A, so we can find P(AC) by finding 1 – P(A) Why do you think this works? Discuss with students that the probabilities of all possible outcomes must add to 1, so the probability of something not happening would be 1 minus the probability of the event occuring.

An experiment consists of tossing three coins. List the sample space for the outcomes of the experiment. {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Find the following probabilities: P(all heads) 1/8 or 12.5% b. P(two tails) 3/8 or 37.5% P(no heads) 1/8 or 12.5% P(at least one tail) 7/8 or 87.5% How could you use compliments to find d? The compliment of at least one tail is no tails, so you could do 1 – P(no tails) = 1 – 1/8 = 7/8 or 87.5%

A bag contains six red marbles, four blue marbles, two yellow marbles and 3 white marbles. One marble is drawn at random. List the sample space for this experiment. {r, r, r, r, r, r, b, b, b, b, y, y, w, w, w} Find the following probabilities: a. P(red) 2/5 or 40% b. P(blue or white) 7/15 or 47% c. P(not yellow) 13/15 or 87% (Note that we could either count all the outcomes that are not yellow or we could think of this as being 1 – P(yellow). Why is this?)

Given the Venn Diagram below, find the probability of the following if a student was selected at random: 16.) P( blonde hair) 13/26 or ½ or 0.5 or 50% 17.) P(blonde hair and blue eyes) 8/26 or 4/13 or 0.308 or 30.8% 18.) P(blonde hair or blue eyes) 15/26 or 0.577 or 57.7% 19.) P(not blue eyes) 16/26 or 8/13 or 0.615 or 61.5%

Mutually Exclusive Events Suppose you are rolling a six-sided die. What is the probability that you roll an odd number or you roll a 2? Can these both occur at the same time? Why or why not? Mutually Exclusive Events (or Disjoint Events): Two or more events that cannot occur at the same time. The probability of two mutually exclusive events occurring at the same time , P(A and B), is 0! Video on Mutually Exclusive Events

Probability of Mutually Exclusive Events To find the probability of one of two mutually exclusive events occurring, use the following formula: P(A or B) = P(A) + P(B) Here’s an easy way to remember: AND in a probability problem with two events means to MULTIPLY (this is what we learned last class!) OR in a probability problem with two events means to ADD

Examples If you randomly chose one of the integers 1 – 10, what is the probability of choosing either an odd number or an even number? Are these mutually exclusive events? Why or why not? Complete the following statement: P(odd or even) = P(_____) + P(_____) P(odd or even) = P(odd) + P(even) Now fill in with numbers: P(odd or even) = _______ + ________ P(odd or even) = ½ + ½ = 1 Does this answer make sense?

2. Two fair dice are rolled 2. Two fair dice are rolled. What is the probability of getting a sum less than 7 or a sum equal to 10? Are these events mutually exclusive? Sometimes using a table of outcomes is useful. Complete the following table using the sums of two dice: Die 1 2 3 4 5 6 7

Die 1 2 3 4 5 6 7 8 9 10 11 12 P(getting a sum less than 7 OR sum of 10) = P(sum less than 7) + P(sum of 10) = 15/36 + 3/36 = 18/36 = ½ The probability of rolling a sum less than 7 or a sum of 10 is ½ or 50%. You can do the activity at this point or at the end of the examples.

Mutually Inclusive Events Suppose you are rolling a six-sided die. What is the probability that you roll an odd number or a number less than 4? Can these both occur at the same time? If so, when? Mutually Inclusive Events: Two events that can occur at the same time. Video on Mutually Inclusive Events

Probability of the Union of Two Events: The Addition Rule We just saw that the formula for finding the probability of two mutually inclusive events can also be used for mutually exclusive events, so let’s think of it as the formula for finding the probability of the union of two events or the Addition Rule: P(A or B) = P(A) + P(B) – P(A and B) ***Use this for both Mutually Exclusive and Inclusive events***

Examples If event A: choosing a club from a deck of cards and event B: choosing a 10 from a deck of cards, what is the P(A OR B)? Here is the Venn Diagram of the situation. P(choosing a club or a ten) = P(club) + P(ten) – P(10 of clubs) = 13/52 + 4/52 – 1/52 = 16/52 = 4/13 or .308 The probability of choosing a club or a ten is 4/13 or 30.8% B A A and B

= P(<5) + P(odd) – P(<5 and odd) 2. What is the probability of choosing a number from 1 to 10 that is less than 5 or odd? (Draw a Venn Diagram of the situation first) P(<5 or odd) = P(<5) + P(odd) – P(<5 and odd) <5 = {1,2,3,4} odd = {1,3,5,7,9} = 4/10 + 5/10 – 2/10 = 7/10 The probability of choosing a number less than 5 or an odd number is 7/10 or 70%. < 5 Odd 6 2 5 7 8 1 4 3 10 9 Less than 5 and Odd

3. A bag contains 26 tiles with a letter on each, one tile for each letter of the alphabet. What is the probability of reaching into the bag and randomly choosing a tile with one of the first 10 letters of the alphabet on it or randomly choosing a tile with a vowel on it? Before you try the formula, see if you can figure it out logically! Now, let’s check with the formula P(one of the first 10 letters or vowel) = P(one of the first 10 letters) + P(vowel) – P(first 10 and vowel) = 10/26 + 5/26 – 3/26 = 12/26 or 6/13 The probability of choosing either one of the first 10 letters or a vowel is 6/13 or 46.2%

4. A bag contains 26 tiles with a letter on each, one tile for each letter of the alphabet. What is the probability of reaching into the bag and randomly choosing a tile with one of the last 5 letters of the alphabet on it or randomly choosing a tile with a vowel on it? Before you try the formula, see if you can figure it out logically! Now, let’s check with the formula P(one of the last 5 letters or vowel) = P(one of the last 5 letters) + P(vowel) – P(last 5 and vowel) = 5/26 + 5/26 – 0 = 10/26 or 5/13 The probability of choosing either one of the first 10 letters or a vowel is 5/13 or 38.5%

b) Are A and B mutually exclusive? No, because P(A and B) = 0.3 (≠ 0 ) 5. P(A) = 0.7, P(B) = 0.4, P(A and B) = 0.3 Make a Venn Diagram representing the situation. b) Are A and B mutually exclusive? No, because P(A and B) = 0.3 (≠ 0 ) c) Are A and B independent events? No, because if they were independent, then P(A and B) = P(A) • P(B), and 0.3 ≠ 0.7 • 0.4 d) Find P(A or B) P(A or B) = P(A) + P(B) – P(A and B) P(A or B) = 0.7 + 0.4 – 0.3 P(A or B) = 0.8 or 80% e.) EXTENSION: Find the following probabilities: P (A)C P(A and B)C P(A or B)C = 0.6 = 0.7 = 0.2 A B 0.7 – 0.3= 0.4 0.4 – 0.3 = 0.1 0.3