MDM 4U Probability Review Properties of Probability Experimental Probability - probability measured by performing an experiment for a number of n trials and recording the number of outcomes Theoretical Probability - probability measured by using counting techniques to determine the total number of possible outcomes in an experiment Outcomes - the possible results of an experiment Event - the outcome(s) of interest in the experiment The probability of an event must always be assigned a number between 0 (impossible) and 1 (certain). The general formula for probability for a given event A is: P(A) = Number of outcomes that make up an event A Total Number of possible outcomes This can be written as where n(a) is the number of outcomes that make up event A, and n(s) is the total number outcomes possible for the experiment. The complement of a probability (A ) is given by : Find the probability that a number picked at random between 1 and 10 is (a) even (b) odd
Mutually Exclusive Events If two events have no outcomes in common they are mutually exclusive (ie. The two sets are disjoint). if A, B are mutually exclusive. If two events share outcomes, they are not mutually exclusive. if A, B are not mutually exclusive. The probability that a teenager enjoys rock music is 0.7 and the probability that a teenager enjoys country and western music is 0.2. If only one in ten teenagers enjoys rock as well as country music, what is the probability that a teenager chosen at random listens to either rock or country western? Conditional Probability If the probability of one event depends on the likelihood a separate event occurring we have a conditional probability. where the notation reads the probability of event A given that event B has occurred. What is the probability of rolling a sum greater than 7 with two dice if it is known that first die rolled is a 3? Independent Events If two events have no effect on one another than they are said to be independent. if A and B are independent events. What is the probability of rolling a six with a die and drawing an ace from a deck of cards?
Probability Distributions - the probability of each possible outcome displayed in tabular or graphical form Determine the probability distribution for the possible sums when rolling two dice. Solution OR To construct a probability model it is necessary to assign a numerical value to each possible outcome. The assignment of a numerical value to a real-life occurrence is called the random variable and is usually denoted by X.
Expected Value of a Probability Distribution The expected value is that quantity that you can expect to obtain when the experiment is performed. where X is the random variable and x represents all the possible numeric values which can be assigned for X. All probability distributions are described by: A committee of four people is to be chosen randomly from four males and six females. What is the expected number of females on the committee? Uniform Distributions The simplest probability distribution is one where each possible outcome is equally likely. This type of distribution is called uniform. The simplest example of a uniform distribution is flipping a coin. where 0 = heads, 1 = tails The probability of a given outcome x for P(X) is: where n is the number of outcomes possible for the experiment The Binomial Distribution A stochastic process is referred to as repeated trials if: a) the experiments are identical b) the experiments are independent When evaluating experiments that involve independent, repeated trials where we are only interested in the frequency of the occurrence of one single event we can use the Binomial Distribution.
The event of interest is considered a success. All other events are then considered to be failures. This success/failure analyses taken for repeated identical trials are called Bernoulli trials. The probability distribution of the number of successes in a sequence of Bernoulli trials is called the binomial distribution. where n is the total number of trials, x is the number of successes, p is the probability of success and q is the probability of failure. Remember: A pair of dice are rolled 20 times. What is the probability that double sixes are rolled exactly twice? The Expected Value of the Binomial Distribution The expected number of successes in a binomial distribution with the probability p of success is given by: Waiting Times in a Binomial Distribution The expected number of trials performed before the first success is given by: The probability of success on a given trial is given by: where x = 0, 1, 2... Remember: To apply a binomial distribution: 1. The outcome of each trial can be described by either a success or failure. 2. The trials are identical. 3. The trials are independent.