Probability definitions and properties

This presentation includes the definitions and properties of basic terms in Probability.

1. PROBABILITY DEFINITIONS AND PROPERTIES BY: Razi Ali Valliani 2012222 Shan Virani 2012228 Hamza Qureshi 2012214 Ritick Kumar 2012224

2. What is probability? Probability is the way of expressing knowledge of belief that an event will occur in chance. Probability is the measure of how likely it is the some event will occur, a number expressing the ratio of favorable cases to the whole number or cases possible. Formula: no.of favorable outcomes/Total no.of events What is probability?

3. Example Consider the example of finding the probability of selecting a black card or a 6 from a deck of 52 cards. Solution: We need to find out P(B or 6) Probability of selecting a black card = 26/52 Probability of selecting a 6 = 4/52 Probability of selecting both a black card and a 6 = 2/52 P(B or 6) = P(B) + P(6) – P(B and 6) = 26/52 + 4/52 – 2/52 = 28/52= 7/13.

4. Where is probability used? Probability is used a lot in daily life. Probability is used in such as math, statistics, Finance, Gambling, Science, Machines and artificial intelligence, and in many other activities.

5. How do we express probabilities? Mostly, we express probabilities as fractions. The numerator shows possible number of ways an event can occur. The denominator is a total number of possible event that could occur.

6. Some common terms related to probability. Experiment: Is a situation involving chance or probability that leads to results called outcomes. Outcome: A possible result of a random experiment. Sample space: The set of outcomes of an experiment is known as sample space. Event: One or more outcomes in an experiment. Sample point: Each element of sample space is called a sample point.

7. General rules of probability. Probability is a number between 0 and 1. The sum of probabilities of all possible outcomes in a sample space is 1. The probability that an event does not occur is 1 minus the probability that it does occur.(also called component of A).

8. Independent Events Two events are independent if outcome of event A, has no effect on outcome of event B. When multiple events occur, if the outcome of one event DOES NOT affect the outcome of the other events, they are called independent events. For Example: Say, a die is rolled twice. The outcome of the first roll doesn’t affect the second outcome. These two are independent events.

9. Example 1 Example 1: Say, a coin is tossed twice. What is the probability of getting two consecutive tails ? Probability of getting a tail in one toss = ½ The coin is tossed twice. So 1/2 * 1/2 = 1/4 is the answer. Here’s the verification of the above answer with the help of sample space. When a coin is tossed twice, the sample space is {(H,H), (H,T), (T,H), (T,T)}. Our desired event is (T,T) whose occurrence is only once out of four possible outcomes and hence, our answer is 1/4.

10. Example 2: Consider another example where a pack contains 4 blue, 2 red and 3 black pens. If a pen is drawn at random from the pack, replaced and the process repeated 2 more times, What is the probability of drawing 2 blue pens and 1 black pen ? Solution Here, total number of pens = 9 Probability of drawing 1 blue pen = 4/9 Probability of drawing another blue pen = 4/9 Probability of drawing 1 black pen = 3/9 Probability of drawing 2 blue pens and 1 black pen = 4/9 * 4/9 * 3/9 = 48/729 = 16/243 Example 2

11. Conditional probability The conditional probability of an event A is the probability that an event A will occur given that another event, B, has occurred. Formula: probability(A|B)=P(AnB)/P(B).

12. Example In a class, 40% of the students study math and science. 60% of the students study math. What is the probability of a student studying science given he/she is already studying math ? Solution P(M and S) = 0.40 P(M) = 0.60 P(S|M) = P(M and S)/P(S) = 0.40/0.60 = 2/3 = 0.67

13. Experimental probability An experimental probability is one that happens as the result of an experiment. Formula: (# of outcomes)/(# of trials) Example: Weather Forecasting. Before planning for an outing or a picnic, we always check the weather forecast. ...