What is Probability and its Basic Definitions
- 1. Probability By Dr Chandhini J Assistant Professor of Mathematics, Sri Ramakrishna College of Arts & Science, Coimbatore.
- 2. Why Learn Probability? Nothing in life is certain. In everything we do, we gauge the chances of successful outcomes, from business to medicine to the weather A probability provides a quantitative description of the chances or likelihoods associated with various outcomes It provides a bridge between descriptive and inferential statistics Population Sample Probability Statistics
- 3. What is Probability? We use graphs and numerical measures to describe data sets which were usually samples. We measure “how often” using Relative frequency = f/n Sample And “How often” = Relative frequency Population Probability • As n gets larger,
- 4. Basic Concepts An experiment is the process by which an observation (or measurement) is obtained. An event is an outcome of an experiment, usually denoted by a capital letter. The basic element to which probability is applied When an experiment is performed, a particular event either happens, or it doesn’t!
- 5. Experiments and Events Experiment: Record an age A: person is 30 years old B: person is older than 65 Experiment: Toss a die A: observe an odd number B: observe a number greater than 2
- 6. Basic Concepts Two events are mutually exclusive if, when one event occurs, the other cannot, and vice versa. •Experiment: Toss a die –A: observe an odd number –B: observe a number greater than 2 –C: observe a 6 –D: observe a 3 Not Mutually Exclusive Mutually Exclusive B and C? B and D?
- 7. Basic Concepts An event that cannot be decomposed is called a simple event. Denoted by E with a subscript. Each simple event will be assigned a probability, measuring “how often” it occurs. The set of all simple events of an experiment is called the sample space, S.
- 8. Example The die toss: Simple events: Sample space: 1 2 3 4 5 6 E1 E2 E3 E4 E5 E6 S ={E1, E2, E3, E4, E5, E6} S •E1 •E6 •E2 •E3 •E4 •E5
- 9. Basic Concepts An event is a collection of one or more simple events. •The die toss: –A: an odd number –B: a number > 2 S A ={E1, E3, E5} B ={E3, E4, E5, E6} B A •E1 •E6 •E2 •E3 •E4 •E5
- 10. The Probability of an Event The probability of an event A measures “how often” A will occur. We write P(A). Suppose that an experiment is performed n times. The relative frequency for an event A is n f n = occurs A times of Number n f A P n lim ) ( → = • If we let n get infinitely large,
- 11. The Probability of an Event P(A) must be between 0 and 1. If event A can never occur, P(A) = 0. If event A always occurs when the experiment is performed, P(A) =1. The sum of the probabilities for all simple events in S equals 1. • The probability of an event A is found by adding the probabilities of all the simple events contained in A.
- 12. – Suppose that 10% of the U.S. population has red hair. Then for a person selected at random, Finding Probabilities Probabilities can be found using Estimates from empirical studies Common sense estimates based on equally likely events. P(Head) = 1/2 P(Red hair) = .10 • Examples: –Toss a fair coin.
- 13. Using Simple Events The probability of an event A is equal to the sum of the probabilities of the simple events contained in A If the simple events in an experiment are equally likely, you can calculate events simple of number total A in events simple of number ) ( = = N n A P A
- 14. Example 1 Toss a fair coin twice. What is the probability of observing at least one head? H 1st Coin 2nd Coin Ei P(Ei) H T T H T HH HT TH TT 1/4 1/4 1/4 1/4 P(at least 1 head) = P(E1) + P(E2) + P(E3) = 1/4 + 1/4 + 1/4 = 3/4
- 15. Example 2 A bowl contains three M&Ms, one red, one blue and one green. A child selects two M&Ms at random. What is the probability that at least one is red? 1st M&M 2nd M&M Ei P(Ei) RB RG BR BG 1/6 1/6 1/6 1/6 1/6 1/6 P(at least 1 red) = P(RB) + P(BR)+ P(RG) + P(GR) = 4/6 = 2/3 m m m m m m m m m GB GR
- 16. Example 3 The sample space of throwing a pair of dice is
- 17. Example 3 Event Simple events Probability Dice add to 3 (1,2),(2,1) 2/36 Dice add to 6 (1,5),(2,4),(3,3), (4,2),(5,1) 5/36 Red die show 1 (1,1),(1,2),(1,3), (1,4),(1,5),(1,6) 6/36 Green die show 1 (1,1),(2,1),(3,1), (4,1),(5,1),(6,1) 6/36
- 18. Counting Rules Sample space of throwing 3 dice has 216 entries, sample space of throwing 4 dice has 1296 entries, … At some point, we have to stop listing and start thinking … We need some counting rules
- 19. The mn Rule If an experiment is performed in two stages, with m ways to accomplish the first stage and n ways to accomplish the second stage, then there are mn ways to accomplish the experiment. This rule is easily extended to k stages, with the number of ways equal to n1 n2 n3 … nk Example: Toss two coins. The total number of simple events is: 2 2 = 4
- 20. Examples Example: Toss three coins. The total number of simple events is: 2 2 2 = 8 Example: Two M&Ms are drawn from a dish containing two red and two blue candies. The total number of simple events is: 6 6 = 36 Example: Toss two dice. The total number of simple events is: m m 4 3 = 12 Example: Toss three dice. The total number of simple events is: 6 6 6 = 216