Probability Notes: Binomial Probabilities A Binomial Probability is a type of discrete probability with only two outcomes (tea or coffee, win or lose, have disease or don t have disease). The category you are finding the percentage of is the success category or in Statcato it is called the event probability. Success Category (have disease) Failure Category (don t have disease) For a binomial probability, individual observations should be independent of each other with a consistent probability of success. (For example, winning at cards often fails this assumption because the number of cards and the probabilities are always changing.) Calculating Binomial Probabilities When calculating binomial probabilities, you need three bits of information. Number of events (successes) (X) Event probability (probability that the success category happens 1 time. (p) Total number of trials (total number of people or total number of times you plan to play the game, sample size) (n) The Binomial formula programmed into computers is this. This calculates only the P(x =#) and has to be repeated over and over to calculate less than or greater than. Lucky computers can handle the computations. PP(XX) = CC(nn, xx) pp xx (1 pp) nn xx
Calculate Binomial Probabilities with Statcato Calculate Menu => Probability Distributions => Binomial Enter the total number of people or times played under number of trials Enter the % (as a decimal proportion) for one success under event probability. Enter the number of success (X) under constant Calculate Binomial Probabilities with StatCrunch Stat Menu => Calculator => Binomial => Standard or Between
Note about inequality symbols. Normal Probabilities: When dealing with continuous quantitative data with decimals, we had infinite totals so the probability of less than 3 kilograms is 2.999999999 or below. Hence for normal probabilities the probability of less than 3 is about the same as less than or equal to. Binomial Probabilities: This is not the case for binomial probabilities. Winning a game less than 3 times means winning less than or equal to 2 times. So be careful about the wording with inequalities. For Binomial calculations in Statcato, probability density finds the % for number of events equal to a #, cumulative probability finds the % for the number of events less than or equal to a #. Remember subtracting the cumulative probability from 100% will give the % for strictly greater than. For Binomial calculations in StatCrunch, you have the options of =, <, >,, Remember greater than points right and less than points left.
Wording examples = probability that exactly 5 people have the disease (In statcato, calculate with the probability density function with 5 events.) > probability that she wins more than 4 times (Notice more than 4 means greater than or equal to 5. In Statcato, calculate less than or equal to 4 with the cumulative probability function and subtract the answer from 100%) probability that she wins 4 or more times or at least 4 (In Statcato, calculate less than or equal to 3 with the cumulative probability function and subtract the answer from 100%) < probability that he wins less than 6 times (notice less than 6 means less than or equal to 5. In Statcato, calculate less than or equal to 5 with the cumulative probability function) probability that he wins 6 times or less or at most 6 (In Statcato, calculate less than or equal to 6 with the cumulative probability function)
Let s look at some examples. Sarah likes to play slot machines in a Casino in Las Vegas. The particular slot machine she is playing has a 7% chance of winning. Suppose Sarah plays the game 35 total times. 1. What is the probability that Sarah wins exactly 2 times. In Statcato, enter 35 under number of trials, 0.07 under event probability and 2 under constant. We will use the probability density button since we are calculating equal to. So the answer is 26.6%
2. What is the probability that Sarah wins more than 3 times? First notice that more than 3 means 4 or more. The opposite of 4 or more is 3 or less. So we will use the cumulative probability button to calculate 3 or less in Statcato. Then subtract the answer from 100%. In Statcato, enter 35 under number of trials, 0.07 under event probability and 3 under constant. Don t forget this calculated the percentage of 3 or less wins not 4 or more. So subtract the answer from 100%. Answer: 100% - 77.3% = 22.7%