PROBABILITY

[Audio] Presentation on probability Presented by K.N.Jeya.

DEFINITION. Probability is a measure of how likely something will occur. It is ratio of desired outcomes to total outcomes. (#desired / #total) Probabilities of all outcomes sums to 1..

EXAMPLE. If I roll number cube, there are six total possibilities.(1,2,3,4,5,6) Each possibility has only one outcome, so each has a PROBABILITY of 1/6. For instance, the probability I roll a 2 is 1/6, since there is only a single 2 on the number cube..

PRACTICE. If I flip a coin, what is the probability I get heads? What is the probability I get tails? Remember, to think of how many possibilities are there!.

ANSWER. P(heads) = ½ P(tails) = ½ If you add these two up, you will get 1, which means the answers are probably right..

TWO OR MORE EVENTS. If there are two or more events, you need to consider if it is happening at the same time or one after the other..

“ AND”. If the two events are happening at the same time, you need to multiply the two probabilities together. Usually, the questions use the word “AND” when describing the outcomes ..

“OR”. If the two events are happening one after the other, you need to add the two probabilities. Usually, the questions use the word “OR” when describing the outcomes..

PRACTICE. If I roll a number cube and flip a coin: What is the probability I will get a heads and a 6? What is the probability I will get a tails or a 3?.

ANSWERS. p(heads and 6) = 1/2 x 1/6 =1/12 P(tails or a 5) = 1/2 + 1/6 = 8/12 =2/3.

EXPERIMENTAL PROBABILITY. An experimental probability is one that happens as the result of an experiment. (# of outcomes) / (# of trials) The probabilities we have done so far are “theoretical probabilities”, because there was no experiments..

SURE OR CERTAIN EVENT. A sure event is an event, which always happens. An event which is sure to occur at every performance of an experiment is called a certain event connected with the experiment. Certain Events also known as Sure Event..

EXAMPLE. For example ,it's a sure event to obtain a number between 1 and 6 when rolling an ordinary die. The probability of a sure event has the value of 1. The probability of an impossible event has the value of 0..

EXAMPLE PROBLEM. The probability of a sure event is Answer: The value of the probability of an impossible event = 0 The value of the probability of a sure event = 1 An event is a sure event that always happens. Example: it’s a sure event to obtain a number between 1 n 6 when rolling an ordinary die..

IMPOSSIBLE EVENT. An event which cannot occur at any performance of the experiment is called an possible event. In other words, An event E is called an impossible event if P(E) = 0. This happens when no outcome of the experiment is a favourable outcome..

EXAMPLE. I n throwing a die, the event of getting a natural number greater than 6 is an impossible event ..

MUTUALLY EXCLUSIVE EVENTS. If there be no element common between two or more events, i.e., between two or more subsets of the sample space, then these events are called mutually exclusive events. If E 1 and E 2 are two mutually exclusive events, then E 1 ∩ E 2 = ∅.

EXAMPLE. I n connection with throw a die “even face” and “odd face” are mutually exclusive. But” odd-face” and “multiple of 3” are not mutually exclusive, because when “face-3” occurs both the events “odd face” and “multiply of 3” are said to be occurred simultaneously. We see that two simple-events are always mutually exclusive while two compound events may or may not mutually exclusive..

EXAMPLE PROBLEM. What is the probability of a die showing a number 3 or number 5? Solution: Let, P(3) is the probability of getting a number 3 P(5) is the probability of getting a number 5 P(3) = 1/6 and P(5) = 1/6 So, P(3 or 5) = P(3) + P(5) P(3 or 5) = (1/6) + (1/6) = 2/6 P(3 or 5) = 1/3 Therefore, the probability of a die showing 3 or 5 is 1/3..

COMPLIMENTARY EVENT. An event which consists in the negation of another event is called complementary event of the er event. In case of throwing a die, ‘even face’ and ‘odd face’ are complementary to each other. “Multiple of 3” ant “Not multiple of 3” are complementary events of each other..

COMPLIMENTARY EVENT. In other words, If E and F are two events for an experiment such that every favorable outcome for the event E is not a favorable outcome for the event F and every unfavorable outcome for the event E is a favorable outcome for F then F is called the complementary event of the event E, and F is denoted by E¯¯¯¯E¯..

EXAMPLE. In the throw of a die if, E = event of getting an odd number then, E¯ = event of not getting an odd number, that is, event of getting an even number..

EXAMPLE PROBLEM. A bag contains 4 red balls and 5 green balls. A ball is drawn from the bag at random. Let E = event of drawing a red ball. Then, E = event of not drawing a red ball = event of drawing a green ball. Now, Number of Outcomes Favourable to E Total Number of Possible Outcomes 4 [Since there are 4 red balls]. Number of Outcomes Favourable toE Total Number of Possible Outcomes 5 [Since there are 5 green balls]. so, P(E) + P(E) = Therefore, P(E) - 5 9 9 -1- P(E) and P(E) -.

CLASSICAL APPROACH. A n approach to the understanding of probability based on the assumptions that any random process has a given set of possible outcomes and that each possible outcome is equally likely to occur..

EXAMPLE. An example often used is rolling a die, in which there are six possible outcomes and each outcome is assumed to be equally likely..

FORMULA. Classical approach==P(A) = f / N. P(A) means “probability of event A” (event A is whatever event you are looking for, like winning the lottery). “f” is the frequency, or number of possible times the event could happen. N is the number of times the event could happen..

EMPIRICAL APPROACH. Empirical probability, also known as experimental probability, refers to a probability that is based on historical data. In other words, empirical probability illustrates the likelihood of an event occurring based on historical data..

FORMULA. Formula for Empirical Probability Number of Times Occurred Empirical Probability = Total No. of Times Experiment Performed.

EXAMPLE. The table below shows a dice thrown three times and the corresponding result. What is the empirical probability of rolling a 4? Experiment Result Empirical Probability 2 5 1 - 0%. The empirical probability of rolling a 4 is 0%..

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AGE WISE PROBABILITY. India: COVID-19 cases by age group 2021 | Statista.

HOPE YOU LEARNED MORE ABOUT PROBABILITY! THANK YOU…!!!.