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KS3 Bitesize: Maths - Probability - Introduction Rules of Probability, Independent Events - Statistics and Probability The study of probability mostly deals with combining different events and studying these events alongside each other. How these different events relate to each other determines the methods and rules to follow when we're studying their probabilities. Events can be pided into two major categories dependent or Independent events. Independent Events When two events are said to be independent of each other, what this means is that the probability that one event occurs in no way affects the probability of the other event occurring. Dependent Events When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event. For example, if you were to draw a two cards from a deck of 52 cards. There are 4 Aces in a deck of 52 cards On your first draw, the probability of getting an ace is given by: If we don't return this card into the deck, the probability of drawing an ace on the second pick is given by Conditional Probability Rules of Probability

Conditional Probability How to handle Dependent Events Life is full of random events! You need to get a "feel" for them to be a smart and successful person. Independent Events Events can be "Independent", meaning each event is not affected by any other events. Example: Tossing a coin. Each toss of a coin is a perfect isolated thing. What it did in the past will not affect the current toss. The chance is simply 1-in-2, or 50%, just like ANY toss of the coin. So each toss is an Independent Event. Dependent Events But events can also be "dependent" ... which means they can be affected by previous events ... Example: Marbles in a Bag 2 blue and 3 red marbles are in a bag. What are the chances of getting a blue marble? The chance is 2 in 5 But after taking one out you change the chances! So the next time: if you got a red marble before, then the chance of a blue marble next is 2 in 4 if you got a blue marble before, then the chance of a blue marble next is 1 in 4 See how the chances change each time? "Replacement" Tree Diagram Check

Probabilty Tree Diagrams We can construct a probability tree diagram to help us solve some probability problems. A probability tree diagram shows all the possible events. The first event is represented by a dot. In this lesson, we will learn how to draw probability tree diagrams for independent events (with replacement) how to draw probability tree diagrams for dependent events (without replacement) Related Topics:More Probability Lessons Example: A bag contains 3 black balls and 5 white balls. a) Construct a probability tree of the problem. b) Calculate the probability that Paul picks: i) two black balls ii) a black ball in his second draw Solution: a) Check that the probabilities in the last column add up to 1. b) i) To find the probability of getting two black balls, first locate the B branch and then follow the second B branch. ii) There are two outcomes where the second ball can be black. Either (B, B) or (W, B) From the probability tree diagram, we get: P(second ball black) = P(B, B) or P(W, B) (ii) both are black.

Stats: Probability Rules "OR" or Unions Mutually Exclusive Events Two events are mutually exclusive if they cannot occur at the same time. Another word that means mutually exclusive is disjoint. If two events are disjoint, then the probability of them both occurring at the same time is 0. Disjoint: P(A and B) = 0 If two events are mutually exclusive, then the probability of either occurring is the sum of the probabilities of each occurring. Specific Addition Rule Only valid when the events are mutually exclusive. P(A or B) = P(A) + P(B) Example 1: Given: P(A) = 0.20, P(B) = 0.70, A and B are disjoint I like to use what's called a joint probability distribution. The values in red are given in the problem. Non-Mutually Exclusive Events In events which aren't mutually exclusive, there is some overlap. General Addition Rule Always valid. P(A or B) = P(A) + P(B) - P(A and B) Example 2: Given P(A) = 0.20, P(B) = 0.70, P(A and B) = 0.15 Interpreting the table Certain things can be determined from the joint probability distribution.