SINGLE RANDOM VARIABLES AND PROBABILITY DISTRIBUTION Probability Trial and Event: Consider an experiment, which though repeated under essential and identical conditions, does not give a unique result but may result in any one of the several possible outcomes. The experiment is known as Trial and the outcome is called Event. E.g. (1) Throwing a dice experiment getting the n’s 1,2,3,4,5,6 (event) (2) Tossing a coin experiment and getting head or tail (event) Probability Introduction Probability is the likelihood or chance of an event occurring. Probability = the number of ways of achieving success the total number of possible outcomes For example, the probability of flipping a coin and it being heads is ½, because there is 1 way of getting a head and the total number of possible outcomes is 2 (a head or tail). We write P(heads) = ½ . The probability of something which is certain to happen is 1. The probability of something which is impossible to happen is 0. The probability of something not happening is 1 minus the probability that it will happen. Single Events: Example There are 6 beads in a bag, 3 are red, 2 are yellow and 1 is blue. What is the probability of picking a yellow? The probability is the number of yellows in the bag divided by the total number of balls, i.e. 2/6 = 1/3. Example There is a bag full of colored balls, red, blue, green and orange. Balls are picked out and replaced. John did this 1000 times and obtained the following results: Number of blue balls picked out: 300 Number of red balls: 200 Number of green balls: 450 Number of orange balls: 50 a) What is the probability of picking a green ball? For every 1000 balls picked out, 450 are green. Therefore P(green) = 450/1000 = 0.45 b) If there are 100 balls in the bag, how many of them are likely to be green? The experiment suggests that 450 out of 1000 balls are green. Therefore, out of 100 balls, 45 are green (using ratios). Multiple Events Independent and Dependent Events Suppose now we consider the probability of 2 events happening. For example, we might throw 2 dice and consider the probability that both are 6's. We call two events independent if the outcome of one of the events doesn't affect the outcome of another. For example, if we throw two dice, the probability of getting a 6 on the second die is the same, no matter what we get with the first one- it's still 1/6. On the other hand, suppose we have a bag containing 2 red and 2 blue balls. If we pick 2 balls out of the bag, the probability that the second is blue depends upon what the color of the first ball picked was. If the first ball was blue, there will be 1 blue and 2 red balls in the bag when we pick the second ball. So the probability of getting a blue is 1/3. However, if the first ball was red, there will be 1 red and 2 blue balls left so the probability the second ball is blue is 2/3. When the probability of one event depends on another, the events are dependent. Exhaustive Events: The total no. of possible outcomes in any trial is called exhaustive event. E.g.: (1) In tossing of a coin experiment there are two exhaustive events. (2) In throwing an n-dice experiment, there are 6n exhaustive events. Favorable event: The no of cases favorable to an event in a trial is the no of outcomes which entities the happening of the event. E.g. (1) In tossing a coin, there is one and only one favorable case to get either head or tail. Mutually exclusive Event: If two or more of them cannot happen simultaneously in the same trial then the event are called mutually exclusive event. E.g. In throwing a dice experiment, the events 1,2,3, ------6 are M.E. events Equally likely Events: Outcomes of events are said to be equally likely if there is no reason for one to be preferred over other. E.g. tossing a coin. Chance of getting 1,2,3,4,5,6 is equally likely. Independent Event: Several events are said to be independent if the happening or the non-happening of the event is not affected by the concerning of the occurrence of any one of the remaining events. An event that always happen is called Certain event, it is denoted by “S‟. An event that never happens is called Impossible event, it is denoted by “ “. Eg: In tossing a coin and throwing a die, getting head or tail is independent of getting no’s 1 or 2 or 3 or 4 or 5 or 6.