Umm Al-Qura University بسم الله الرحمن الرحيم Umm Al-Qura University Health Sciences College at Al-Leith Department of Public Health Lecture (4)

Probability Theory

Objectives: 1/ Define basics of probability theory. 2/ Give an Example of probability theory.

Probability Definition: Probability The probability of an event is the proportion of times that the event occurs in a large number of trials of the experiment. It is the “long-run relative frequency of the event.” statistic

• Experiment: Draw a card from a standard deck of 52. Definition • Experiment: Draw a card from a standard deck of 52. • Sample space: The set of all possible distinct outcomes, S (e.g., 52 cards). • Elementary event or sample point: a member of the sample space. (e.g., the ace of hearts). • Event (or event class): any set of elementary events. e.g., Suit (Hearts), Color (Red), or Number (Ace). statistic

Example statistic

Formal Definition of Probability: Probability is a number assigned to each and every member in the sample space. Denote by P(・). A probability function is a rule of correspondence that associates with each event A in the sample space S a number P(A) such that • 0 ≤ P(A) ≤ 1, for any event A. • The sum of probabilities for all distinct events is 1. • If A and B are mutually exclusive events, then P(A or B) = P(A ∪ B) = P(A) + P(B) statistic

Example: Let A = number card (i. e. , 2–10), B = face card (i. e Example: Let A = number card (i.e., 2–10), B = face card (i.e., J, Q, K), and C = Ace. Probabilities of events: P(A) = 9(4)/52 = 36/52 = 0.6923 P(B) = 3(4)/52 = 12/52 =0 .2308 P(C) = 1(4)/52 = 4/52 = 0.0769 • P(A) + P(B) + P(C) = 1 • P(A ∪ B) = P(A) + P(B) = 0.6923 + 0.2308 =0.9231 = 48/52. statistic

statistic

statistic

When a coin is tossed, there are two possible outcomes: heads (H) or Tossing a Coin  When a coin is tossed, there are two possible outcomes: heads (H) or text (T) We say that the probability of the coin landing H is ½. And the probability of the coin landing T is ½. statistic

The probability of any one of them is 1/6. Throwing Dice  When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6. The probability of any one of them is 1/6. statistic

Example: the chances of rolling a "4" with a die Number of ways it can happen: 1 (there is only 1 face with a "4" on it) Total number of outcomes: 6 (there are 6 faces altogether) So the probability =   1 6 statistic

Number of ways it can happen: 4 (there are 4 blues) Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked? Number of ways it can happen: 4 (there are 4 blues) Total number of outcomes: 5 (there are 5 marbles in total) So the probability =   4   = 0.8 5 statistic

Example: toss a coin 100 times, how many Heads will come up? Probability says that heads have a ½ chance, so we can expect 50 Heads. But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50. statistic

Example: choosing a card from a deck There are 52 cards in a deck (not including Jokers) So the Sample Space is all 52 possible cards: {Ace of Hearts, 2 of Hearts, etc... } statistic

Each time Alex throws the 2 dice is an Experiment. Example: Alex wants to see how many times a "double" comes up when throwing 2 dice. Each time Alex throws the 2 dice is an Experiment. It is an Experiment because the result is uncertain. The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points: {1,1} {2,2} {3,3} {4,4} {5,5} and {6,6} The Sample Space is all possible outcomes (36 Sample Points): {1,1} {1,2} {1,3} {1,4} ... {6,3} {6,4} {6,5} {6,6} statistic

Exercise 8 The diagram shows a spinner made up of a piece of card in the shape of a regular pentagon, with a toothpick pushed through its center. The five triangles are numbered from 1 to 5. The spinner is spun until it lands on one of the five edges of the pentagon. What is the probability that the number it lands on is odd? 3/5 statistic

Each of the letters of the word MISSISSIPPI are written on separate pieces of paper that are then folded, put in a hat, and mixed thoroughly.  One piece of paper is chosen (without looking) from the hat. What is the probability it is an I? 4/11 statistic

A card is chosen at random from a deck of 52 playing cards A card is chosen at random from a deck of 52 playing cards.  There are 4 Queens and 4 Kings in a deck of playing cards. What is the probability the card chosen is a Queen or a King? statistic 52/4 +8/4

A committee of three is chosen from five councilors - Adams, Burke, Cobb, Dilby and Evans. What is the probability Burke is on the committee? But all statistic

There are 10 counters in a bag: 3 are red, 2 are blue and 5 are green There are 10 counters in a bag: 3 are red, 2 are blue and 5 are green.  The contents of the bag are shaken before Maxine randomly chooses one counter from the bag.  What is the probability that she doesn't pick a red counter? statistic

Sarah goes to her local pizza parlor and orders a pizza Sarah goes to her local pizza parlor and orders a pizza. She can choose either a large or a medium pizza, has a choice of seven different toppings, and can have three different choices of crust.  12 statistic

Trees chart (tree diagram) statistic

Example : You are buying a new car How many total choices? statistic

You can count the choices, or just do the simple calculation: Total Choices = 2 × 5 × 3 = 30 statistic

Example: You are buying a new car ... but ... the salesman says "You can't choose black for the hatchback" ... well then things change! statistic

You now have only 27 choices Because your choices are not independent of each other. But you can still make your life easier with this calculation: Choices = 5×3 + 4×3 = 15 + 12 = 27 statistic

Ben can take any one of three routes from school (S) to the town center (T), and can then take five possible routes from the town center to his home (H). He doesn't retrace his steps. How many different possible ways can Ben walk home from school? 8 statistic

Example: You sell sandwiches. 70% of people choose chicken, the rest choose pork. What is the probability of selling 2 chicken sandwiches to the next 3 customers? statistic

The "Two Chicken" cases are highlighted. 0.147 + 0.147 + 0.147 = 0.441 statistic

Example : 3 boxes Box A contain 10 led light but 4 it bad . Box B contain 6 led light but 1 it bad . Box C contain 10 led light but 3 it bad . What is the probability the chosen is a one led light but bad ? And What is the probability the chosen is a one led light bad from box B? statistic

1 3 1 3 1 3 A C B 6 10 3 8 4 10 5 6 1 6 5 8 P (bad) = 𝟏 𝟑 ( 𝟒 𝟏𝟎 ) + 𝟏 𝟑 ( 𝟏 𝟔 ) + 𝟏 𝟑 ( 𝟑 𝟖 ) = 0.3 P (good) = 𝟏 𝟑 ( 𝟔 𝟏𝟎 ) + 𝟏 𝟑 ( 𝟓 𝟔 ) + 𝟏 𝟑 ( 𝟓 𝟖 ) = 0.7 OR q=1-p = 1- 0.3 = 0.7 P( box B but bad )= 𝟏 𝟑 ( 𝟏 𝟔 ) = 0.05 statistic

Probability : A box of five balls pulled a ball randomly and then restored. Calculate the sample space? sample space = 5*5 = 25 1 2 3 4 5 (1,1) (1,2) (1,3) (1,4) (1,5) (2,1) (2,2) (2,3) (2,4) (2,5) 3 (3,1) (3,2) (3,3) (3,4) (3,5) = 25 pairs 4 (4,1) (4,2) (4,3) (4,4) (4,5) 5 (5,1) (5,2) (5,3) (5,4) (5,5) statistic

if the probability of occurrence of the event A = 0 if the probability of occurrence of the event A = 0.3 and the occurrence of the event A and B = 0.4 What is the probability event occurs B ? P (A,B)= P(A) + P(B) 0.4= 0.3 + P(B) P(B)= 0.4 - .03 = 0.1 statistic

Example : Ball pulled randomly from a unbiased box contains six red balls and four white balls and five blue. 1- Find the sample space . 2- what is the probability that be red ? 3- what is the probability that be white ? 4- what is the probability that be blue ? 5- what is the possibility that is not blue? statistic

OR p( not red ) = p(white + blue) = p (white) + p(blue) = 𝟑 𝟓 Sample space = 6+4+5 = 15 P (red) = 𝟐 𝟓 P(white) = 𝟒 𝟏𝟓 P(blue) = 𝟏 𝟑 P( not red)= 1- p (red) = 𝟑 𝟓 OR p( not red ) = p(white + blue) = p (white) + p(blue) = 𝟑 𝟓 P(red and blue ) = p (red) + p(blue) = 𝟐 𝟑 statistic

Factory contains three machines product 50% and 30% and 20% respectively and the damaged 3% and 4% and 5%, respectively. What is the probability of selecting a random sample of factory production to be damaged ? statistic

p(damaged) = 0.5(0.03)+ 0.3(0.04) + 0.2(0.05) = 0.37 statistic

The End statistic