Probability Distributions

Constructing a Probability Distribution Definition: Consists of the values a random variable can assume and the corresponding probabilities of the values. The probabilities are determined either theoretically or by observations. Must meet 2 requirements: #1 The sum of the probabilities of all the events, in the sample space, must be equal to 1. #2 The probability of each event, in the sample space, must be between or equal to 0 and 1.

Prob. Distribution Examples The probability a tutor sees 1,2,3, or 4 students a day is.63,.20,.12, and.05 respectively. The probability of picking a heart, diamond, spade, or club, when picking one card from a deck. A box contains 3 $10 bills, 4 $50 bills, and 2 $100 bills. Construct a probability distribution for the data.

Probability Distribution? Yes or No Determine if the following actually are probability distributions. X1234 P(X) ⅕⅕⅖⅖ X P(X)

Means of Probability Distributions Can not simply use the old formula of finding the sum and dividing by total number of values. This formula would only approximate the true mean. New Steps: #1 Multiply all possible outcomes by the probability of each outcome. #2 Add all answers together. Formula: ∑ [ X ● P (X) ]

Mean Examples The number of pizzas sold in an hour at a local pizzeria is shown in the table, with the corresponding probabilities. Find the mean of the data. Pizzas Sold (X) Probability, P(X)

Mean Examples The number of children living in a household, in Somerville, is shown in the following table. Find the mean of the data. # of Children01234 Probability, P(X)

Variance and Standard Deviation of Probability Distributions Step 1: Find the mean of the data, non rounded answer, and square it. (To be used later) Step 2: Square each possible outcome and multiply the answer to its corresponding probability. Step 3: Find the sum of the answers of Step 2. Step 4: Step 3 answer – Step 1 answer. (Variance) Standard Deviation = √ Variance.

Variance and Standard Deviation Examples Find the variance and standard deviation of the data below. # of Children01234 Probability, P(X) Pizzas Sold (X) Probability, P(X)

Expected Value Calculated in the same way as the mean for a probability distribution. The symbol for expected value is E(X). In games of chance an expected value = 0 means the game is fair. If expected value is positive then the game is in favor of the player. If negative the game is in favor of the house. Represents the average losses or gains that each person can expect to have if they are part of the game.

E(X) Examples Example 1: A local school is selling 500 tickets for a raffle. Each ticket costs $5. The prizes are one $100 prize, one $50 prize and three $25 prizes. Find the expectation if the person buys just one ticket. Example 2: A player rolls a die. If he/she gets a 1 or 6 they win $10. It costs $4 to roll the die. Find the expectation. Example 3: Jenkinson’s boardwalk loses $60,000 a season when it rains too much and makes $350,000 when it is sunny. The probability of too much rain over the summer is 18%. Find the expectation for the profit.

Binomial Experiment A probability experiment that satisfies four conditions: 1: Each trial can have only two outcomes or outcomes that can be reduced to two outcomes. These outcomes can be considered as either success or failure. 2: There must be a fixed number of trials. 3: The outcomes of each trial must be independent of each other. 4: The probability of a success must remain the same for each trial.

Binomial Distribution The outcomes of the binomial experiment lead to a special probability distribution. Some binomial notation: P(S) = probability of success P(F) = probability of failure p = the numerical probability of success q = the numerical probability of failure. ( 1 – p ) n = number of trials X = the number of successes

Binomial Probability Formula In a binomial experiment, the probability of exactly X successes in n trials is:

Binomial Comparison If a couple has 3 children what is the probability they will have exactly 2 girls? Old method: 1: Use a tree diagram and find the sample space. 2: Determine how many outcomes have exactly 2 girls. Binomial method: 1: Use the formula shown before. Where n = 3, X = 2, p = 1/2, q = ( 1 – p) =1/2

Binomial Examples Example 1: A student takes a 6 question, true- false quiz and guesses on each question. Find the probability of passing if the lowest passing grade is at least 4 correct out of 6. Example 2: If someone tosses a coin 8 times find the probability they will toss exactly 5 tails. If 30% of people in a community use the local emergency room at a hospital in one year, find the probability from a sample of 10 people that exactly 3 used the emergency room.

Mean, Variance, and Standard Deviation of the Binomial Distribution Mean = n ● p Variance = n ● p ● q Standard Deviation = √ Variance Example 1: A die is rolled 50 times. Find the mean, variance, and standard deviation for the number of 5’s that will be rolled. Example 2: A study found that 5% of Americans work within 10 miles of their home. If 650 people are selected at random find the mean, variance, and standard deviation of the number of people who work within 10 miles of their home.