Probability is the study of the chance of events happening. A probability can be expressed as a fraction, decimal, or a percent. Experimental Probability

If P(E) = 0, then the event cannot occur. Impossible and Certain If P(E) = 1, then the event must occur. It is impossible. It is certain.

Experimental Probability One way to determine probability is to do an experiment and analyze the results. Trials = Outcomes = the total number of times the experiment is repeated. the different results possible for one trial of the experiment

Go to: – Login: Geometry A Block – P5DR9HT9F Geometry B Block – H86FJZ9QR Geometry E Block – M6BWC7NJ3J – Gizmos: Theoretical and Experimental Probability Geometric Probability Independent and Dependent Events Probability Simulations Simulations

Experiments: Experiment 1: Toss a coin 20 times. In each case, record the number of times you got ‘Heads’. Place results on the board. Experiment 2: Roll a die 20 times In each case record the number of times you got 1, 2, 3, 4, 5, or 6. Place the results on the board.

Experimental  Theoretical The more trials performed in an experiment, then the experimental probability will approximately equal the theoretical probability.

a) Rolling a single die b) Flipping a coin c) Flipping two coins Example: List the outcomes for: Six Possible Outcomes = 1, 2, 3, 4, 5 or 6 Two Possible Outcomes = H or T Four Possible Outcomes = HH, HT, TH or TT Although HT and TH may be the same?

The chance of an event occurring in any trial of the experiment. Theoretical Probability

Example: A ticket is randomly selected from a basket containing 3 green, 4 yellow and 5 blue tickets. Determine the probability of getting: a)a green ticket b)a green or yellow ticket c)an orange ticket d)a green, yellow or blue ticket 3 / 12 or 25% 7 / 12 or 58.3% 4 / 12 or 33.3% 12 / 12 or 100%

Example: A ordinary 6-sided die is rolled once. Determine the chance of: a)getting a 6 b)getting a 1 or 2 c)not getting a 6 d)not getting a 1 or 2 1 / 6 2 / 6 5 / 6 4 / 6

Example: A bag has 20 coins numbered from 1 to 20. A coin is drawn at random and its number is noted. a) P (even) = b) P (divisible by 3) = a) P (divisible by 3 or 5) = 10 / 20 6 / 20 9 / 20

Example: Draw a table of outcomes to display the possible results when two dice are rolled and the scores are summed. Determine the probability that the sum of the dice is ,11,21,31,41,51,6 2 2,12,22,32,42,52,6 3 3,13,23,33,43,53,6 4 4,14,24,34,44,54,6 5 5,15,25,35,45,55,6 6 6,16,26,36,46,56,6 6 / 36

PQ P  Q U P  Q Intersection of P and Q is P  Q Union of P or Q is P  Q Intersection / Union

Independent Events Events where the occurrence of one of the events ______ _____ affect the occurrence of the other event. P(A  B) = P(A and B) = P(A) × P(B) “and” → multiplication does not P(A and B and C) = P(A) × P(B) × P(C)

Example: A coin and a die are tossed simultaneously. Determine the probability of getting heads and a 3.

Experiment: Flip a Coin and Roll a Die P(A)P(B)P(A and B) P(heads and a 4) P(heads and an odd number) P(tails and a number larger than 1) P(tails and a number less than 3)

Example: There are 9 brown boxes and 6 red boxes on a shelf. Anna chooses a box and replaces it. Brian does the same thing. What is the probability that Anna and Brian choose a brown box?

What if Anna choose the box and DID NOT replace it? There are 9 brown boxes and 6 red boxes on a shelf. If Anna chooses brown, P(Brian chooses brown) = P(Anna then Brian choose brown) P(Brian chooses brown) = If Anna chooses red, Then Brian’s event of choosing a box becomes dependent.

Dependent Events Events where the occurrence of one of the events ______ affect the occurrence of the other event. P(A then B) = P(A) × P(B given that A has occurred) does

Example: A box contains 4 red and 2 yellow tickets. Two tickets are randomly selected one by one from the box, without replacement. Find the probability that: (a) both are red (b) the first is red and the second is yellow.

Example: A hat contains tickets with numbers 1, 2, 3, …, 19, 20 printed on them. If 3 tickets are draw from the hat, without replacement, determine the probability that all are prime numbers. Possible = 2, 3, 5, 7, 11, 13, 17, 19

Mutually Exclusive Events A bag of candy contains 12 red candies and 8 yellow candies. Can you select one candy that is both red and yellow?

P(either A or B) = P(A) + P(B) Mutually Exclusive Events

Example: Of the 31 people on a bus tour, 7 were born in Scotland and 5 were born in Wales. a)Are these events mutually exclusive? b)If a person is chosen at random, find the probability that he or she was born in: i.Scotland ii.Wales iii.Scotland or Wales 7 / 31 5 / / 31

P(either A or B) = P(A) + P(B) – P(A and B) Combined Events

Example: 100 people were surveyed: 72 people have had a beach holiday 16 have had a skiing holiday 12 have had both What is the probability that a person chosen has had a beach holiday or a ski holiday?

Laws of Probability TypeDefinitionFormula Mutually Exclusive Events events that cannot happen at the same time P(A ∩ B) = 0 P(A  B) = P(A) + P(B) Combined Eventsevents that can happen at the same time P(A  B) = P(A) + P(B) – P(A∩B) Independent Events occurrence of one event does NOT affect the occurrence of the other P(A ∩ B) = P(A) P(B)

Find p if: a)A and B are mutually exclusive b)A and B are independent P (A) = ½ P (B) = 1/3 and P(A  B) = p Example