The Binomial Probability Theorem

Can you make a guess what the next one would be? A binomial is a polynomial with two terms such as x + a. Often we need to raise a binomial to a power. In this section we'll explore a way to do just that without lengthy multiplication. Can you see a pattern? Can you make a guess what the next one would be? We can easily see the pattern on the x's and the a's. But what about the coefficients? Make a guess and then as we go we'll see how you did.

Let's list all of the coefficients on the x's and the a's and look for a pattern. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 + + + + + + + + + + 1 5 10 10 5 1 Can you guess the next row?

This is good for lower powers but could get very large This is good for lower powers but could get very large. We will introduce some notation to help us and generalise the coefficients with a formula based on what was observed here. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 This is called Pascal's Triangle and would give us the coefficients for a binomial expansion of any power if we extended it far enough.

The Binomial Theorem Find the 5th term of (x + a)12 The x's start out to the nth power and decrease by 1 in power each term. The a's start out to the 0 power and increase by 1 in power each term. The binomial coefficients are found by computing the combination symbol. Also the sum of the powers on a and x is n. Find the 5th term of (x + a)12 1 less than term number 5th term will have a4 (power on a is 1 less than term number) So we'll have x8 (sum of two powers is 12)

Binomial Probability Theorem The binomial theorem is used to calculate the probability for the outcomes of repeated independent and identical trials. If p is the probability of success and q is the probability of failure (q = 1 - p), then the probability of x successes in n trials is: P(x successes) = nCx px qn - x 1. Hockey cards, chosen at random from a set of 20, are given away inside cereal boxes. Stan needs one more card to complete his set so he buys five boxes of cereal. What is the probability that he will complete his set? P(x successes) = nCx px qn - x = 0.2 The probability of Stan completing his set is 20%.

Binomial Distribution - Applications 2. Seven coins are tossed. What is the probability of four tails and three heads? P(x successes) = nCx px qn - x n = 7 (number of trials) P(4 successes) = x = 4 (number of successes) = 0.273 p = (probability of success) The probability of four heads and three tails is 27%. q = (probability of failure) 3. A true-false test has 12 questions. Suppose you guess all 12. What is the probability of exactly seven correct answers? P(x successes) = nCx px qn - x n = 12 (number of trials) x = 7 (number of successes) P(7 successes) = p = (probability of success) = 0.193 The probability of seven correct answers is 19%. q = (probability of failure)

Binomial Distribution - Applications 4. A test consists of 10 multiple choice questions, each with four possible answers. To pass the test, one must answer at least nine questions correctly. Find the probability of passing, if one were to guess the answer for each question. P(x successes) = nCx px qn - x P(x successes) = P(9 successes) + P(10 successes) = 0.000 0296 The probability of passing is 0.003%.

Binomial Distribution - Applications A family has nine children. What is the probability that there is at least one girl? This can be best solved using the compliment, that is, the probability of zero girls: n = 9 (number of trials) P(x successes) = nCx px qn - x x = 0 (number of successes) P(0 successes) = p = (probability of success) = 0.001 95 n = (probability of failure) The probability of zero girls is 0.001 95, therefore the probability of at least one girl is 1 - 0.001 95 = 0.998.

Binomial Distribution - Applications While pitching for the Toronto Blue Jays, 4 of every 7 pitches Juan Guzman threw in the first 5 innings were strikes. What is the probability that 3 of the next 4 pitches will be strikes? Let’s assume that there are only two possible outcomes, strikes or balls. n = 4 (number of pitches) P(x successes) = nCx px qn - x x = 3 (number of strikes) P(3 successes) = p = (probability of success) = 0.32 q = (probability of failure) Note: will also yield the same result.

Using the Binomial Theorem to Calculate Probabilities What is the probability of correctly guessing the outcome of exactly one out of four rolls of a die? . The probability of correctly guessing one roll of the die is The probability of incorrectly guessing the outcome is . The probability of one correct and three incorrect guesses is . The correct guess can occur in any one of the four rolls so there are 4C1 ways of arranging the correct guess. P(one correct guess in four rolls) = = 0.386 NOTE: This experiment is called a binomial experiment because it has two outcomes: guessing correctly guessing incorrectly

Using the Binomial Theorem to Calculate Probabilities Find the probability of correctly guessing the outcome of exactly six out of six rolls of the die. NOTE: This is a binomial experiment because there are only two outcomes for each roll: guessing correctly or incorrectly. For this experiment, let p represent the probability of a correct guess and q represent the probability of an incorrect guess. Use the binomial theorem. Expand and evaluate (p + q)6, where Thus, the probability of correctly guessing the outcome of six out of six rolls is

binompdf (number of trials, probability of success, x-value) Using the Binomial Theorem to Calculate Probabilities [cont’d] You can use the table feature of a graphing calculator to calculate probabilities. The table shows the probability for any number of correct guesses. Using the Binomial Probability Distribution feature of the TI-83: DISTR 0: binompdf binompdf (number of trials, probability of success, x-value) n = 6 p = 0.17 P(6 successes) binompdf(6, 0.17, 6) = 0.000 021 4