3.4 Elements of Probability

Probability helps us to figure out the liklihood of something happening. The “something happening” is called and event. The probability of the occurrence of an event is expressed as a fraction or a decimal from 0 to 1. In any probability problem it’s important to identify all of the outcomes that could occur.

Jar of 10 marbles: 4 Red and 6 Blue Find the probability of drawing a red marble at random. Number of favorable outcomes Total number of possible outcomes (sample space) Draw a red marble = 4/10 Total number of marbles The sum of all possibilities = 1 The probability of not picking a red marble is: 1 – 4/10 or 10/10 – 4/10 = 6/10

Sample Space We have a nickel and a quarter that we toss in the air. Some things we might record: 1. an outcome of the number of heads observed. sample space is A = {0,1,2} 2. the sequence of heads and tails. sample space is A = {HH,HT,TH,TT} 3. When both coins land with both heads, both tails or do not match. sample space is A = {M, N} (M= match, N = No Match)

Sample Space Example Determine the sample space for tossing a six sided die twice. The die has six possibilities for the first throw. The die has six possibilities for the second throw. The sample space A = 6X6 = 36 elements.

Events Determine the events for rolling a die twice A. The sum of the numbers showing is 8 The event is {(2,6), (3,5), (4,4), (5,3), (6,2)} B. The sum of the numbers is showing is at least 10. The event is all ordered pairs where the sum is 10, 11 or 12. The event is {(4,6),(5,5),(5,6),(6,4),(6,5),(6,6)}

Certain Event “A” being the sample space, is also known as the certain event. The empty subset of “A” is called the impossible event. Events are sets and can be combined applying the operations of union, intersection and complementation to form new events. Sample space “A” is the universal set for these events. A = {1,2,3,4,5,6}

We toss a die E is the event that the number is even. F is the event that the number is prime E = {2,4,6} F = {2,3,5} The event that the number is either even or prime is E ∪ F = {2,3,4,5,6} The event that the number is an even prime: E ∩ F = {2} The event the number is not even E = {1,2,5} The event the number is not prime F = {1,4,6}

Assigning probabilities to events Each event E is assigned a number p(E) called the probability of the event of E. Sample space A = {1, 2, 3, 4, 5, 6} p 1 p 2 p 3 p 4 p 5 p 6 Probabilities p 1 = 1/12 E= the outcome is an even number p 2 = 1/12 E = {2,4,6} p 3 = 1/3 p(E) = p 2 + p 4 + p 6 p 4 = 1/6 = 1/12 + 1/6 + 1/12 = 4/12 p 5 = ¼ (1/12 + 2/12 + 1/12) p 6 = 1/12 Sum = 1

A box contains 6 red balls and 4 green balls 4 balls are selected at random. What is the probability that 2 will be red and 2 will be green? Total number of outcomes: 10 C 4 = 10!/4!(10-4)! = 210 Task 1: choose 2 red balls from 6 6 C 2 6!/2!(6-2)! = 15 Task 2: choose 2 green balls from 4 4 C 2 4!/2!(4-2)! = 6 (15)(6) = 90 number of favorable outcomes 90/210 total number of possible outcomes

Array search steps An Array length of 10 is searched for a keyword. The number of steps needed to find it is recorded. Use the formula n+1/2 In this case n = 10 (10 + 1) / 2 = 5.5 On average we can expect to find a keyword in 5.5 steps.