Randomness and Probability Statistics Ch. 6

What is probability? A probability is a numerical measure of the likelihood of the event First, think of some event where the outcome is uncertain. Examples --roll of a die, or the amount of rain that we get tomorrow. In each case, we don't know for sure what will happen.

What is Probability The probability of an event is its long-run relative frequency Empirical Probability ( Experimental Probability): For any event A, P( A)= # times A occurs total # of trials In the long run. Law of Large number Toss a coin 50 times record the number of heads ( we are experimenting)

More on Probability Theoretical Probability: P(A): # outcomes in A # of possible outcomes What is the Theoretical Probability of picking Heads with a toss of a fail coin? P (H)= How many heads on a coin How many total possible outcomes P(H)=1/2 The sample space is S={H,T}

Calculator Simulation Look at calculator for simulation

Smart Exchange Simulation

Another example Toss 2 dice at the same time, record their sum. Repeat this process 50 times. What is the experimental probability that when you toss two dice that their sum will be 7? What is the theoretical probability that your sum will be 7? Show the sample spaces

Sample Space

Tossing dice continue What is the theoretical probability that your sum will be 7? P(7)=6/36 =1/6

Tree Diagram If a coin is tossed and the number cube is rolled simultaneously then the probability of getting head on the coin and the number 4 on the number cube is

Tree Diagram

Tree Diagram Cont. The probability is 1/12. Using the multiplication counting principle _______ times _________ # of choices # of choices for tossing a die picking a coin This gives us the total number of outcomes.

Another tree diagram If two coins are tossed simultaneously then the possible outcomes are 4. The possible outcomes are HH, HT, TH, TT. The tree diagram below shows the possible outcomes.

Picture of tree diagram

Laws of Probability Probability is between 0 and 1 Probability =0 ( if you know it does not occur) Probability =1 ( if you know it does occur) The set of all possible outcomes of a trial must have a probability of 1 The probability of an event occurring is 1 minus the probability that it doesn’t occur. P(A)=1-P(Ac)

Complement of a Event A Example: The P(A)= .3 , then the P (Ac )=? Another example P (Bc )= .65, then P ( B)=? P(B)=.35 Notation Ac , ~A, A`, A with line above it

Addition rules of probability Two events A and B are disjoint ( mutually exclusive) if they have no outcomes in common. P(A or B)= P (A) + P (B)

Example Suppose you roll a die Event A= rolling a 4 on a die Event B = rolling a 5 on a die Are Events A and B disjointed? Yes- these events can not happen at the same time What is the probability of rolling a 4 or 5? P(4 or 5)= 1/6 + 1/6= 2/6=1/3 What is the probability of not rolling a 3? P(~3)= 5/6

Another Example What is the probability of drawing an ace or a king from a deck of cards? P(Ace or King) 4/52 + 4/52= 8/52=2/13

Not Disjointed Events Not disjointed ( Have something in common) P(A or B)= P (A) + P ( B) – P ( A and B) We have to subtract what they have in common. If we do not we will count it twice. Example: When picking a card from a standard deck, what is the probability that you pick a king or a spade? P(k or s)=P(k)+p(s)-P(k and s) 4/52+ 13/52-1/52=16/52 =4/13

Applying the Addition Rule When you get to the light at College and Main, it’s either red , green, or yellow. We know the P(green)=.35 and the P( yellow) =.04. What is the probability the light is red? P (G or Y)= .35 + .04=.39 Then the probability it is red is??? 1-.39=.61 because the sum of the probabilities must equal 1

One more example The American Red Cross says that about 45% of the U.S. population has Type O blood, 40% Type A, 11% Type B, and the rest Type AB. Someone volunteers to give blood. What is the probability that this donor A) has type AB blood .04 B) has type A or type B blood .51 C) is not type o? .55

Multiplication Rule For Independent events A and B, the probability that both A and B occur is the product of the probabilities of the two events P(A and B)=P ( A ) times P(B) provide that A and B are independent. Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring.

Example of Independent Events Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die. Choosing a marble from a jar AND landing on heads after tossing a coin. Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the second card. Rolling a 4 on a single 6-sided die, AND then rolling a 1 on a second roll of the die

Example of Multiplication Rule of Independent Events A coin is tossed and a single 6-sided die is rolled. Find the probability of landing on the head side of the coin and rolling a 3 on the die P(H and 3)= P(H) times P(3) ½ x 1/6= 1/12 A card is chosen at random from a deck of 52 cards. It is then replaced and a second card is chosen. What is the probability of choosing a jack and an eight? P(J and 8)= P ( J) ∙ P (8) 4/52 ∙ 4/52= 16/ 2704=1/169

Example of Multiplication Rule of Independent Events A jar contains 3 red, 5 green, 2 blue and 6 yellow marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. What is the probability of choosing a green and a yellow marble? P (G and Y)= P (G) ∙ P (Y) 5/16 ∙6/16=15/128

Another Example We have determined that the probability that we encounter a green light at the corner of College and Main is .35, a yellow light 0.04, and a red light .61. Let’s think about your morning commute in the week ahead. What is the probability you find the light red both Monday and Tuesday? .3761 What is the probability you do not encounter a red light until Wednesday? .1521

Slot Machine A slot machine has three wheels that spin independently. Each has 10 equally likely symbols: 4 bars, 3 lemons, 2 cherries, and a bell. If you play, what is the probability that You get 3 lemons? 27/1000 or .027 You get 3 bells (the jackpot)? 1/1000 or .001

Slot Machine A slot machine has three wheels that spin independently. Each has 10 equally likely symbols: 4 bars, 3 lemons, 2 cherries, and a bell. If you play, what is the probability that You get no bells? .729 or 729/1000 You get at least one bar ( an automatic loser)? 1- P(no bars)=1-(.60)(.60)(.60)= .784