746 CHAPTER 11 Probability We obtained an answer of , 4 9 the same answer that was obtained using the tree diagram. Note that each of the 3 sets of outcomes has 1 success and 2 failures. Rather than listing all the possibilities containing 1 success and 2 failures, we can use the combination formula to determine the number of possible combinations of 1 success in 3 trials. To do so, evaluate C . 3 1 C 3! (3 1)!1! 3 2 1 2 1 1 3 3 1 = − = ⋅ ⋅ ⋅ ⋅ = Thus, we see that there are 3 ways the 1 success could occur in 3 trials. To compute the probability of 1 success in 3 trials, we can multiply the probability of success in any one trial, p q ,2 ⋅ by the number of ways the 1 success can be arranged among the 3 trials, C . 3 1 Thus, the probability of selecting 1 red ball, P(1), in 3 trials may be determined as follows. P C p q (1) ( ) 3 1 3 2 3 12 27 4 9 3 1 1 2 1 2 = = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ = = The binomial probability formula, which we introduce shortly, explains how to obtain expressions like P C p q (1) ( ) 3 1 1 2 = and is very useful in determining certain types of probabilities. To use the binomial probability formula, the following three conditions must hold. Number of trials Number of successes To Use the Binomial Probability Formula 1. There are n repeated independent trials. 2. Each trial has two possible outcomes, success and failure. 3. For each trial, the probability of success (and failure) remains the same. Before going further, let’s discuss why we can use the binomial probability formula to determine the probability of selecting a specific number of red balls when three balls are selected with replacement. First, since each trial is performed with replacement, the three trials are independent of each other. Second, we may consider selecting a red ball as success and selecting any ball of another color as failure. Third, for each selection, the probability of success (selecting a red ball) is 1 3 and the probability of failure (selecting a ball of another color) is . 2 3 Now let’s discuss the binomial probability formula. Binomial Probability Formula The probability of obtaining exactly x successes, P x( ), in n independent trials is given by the binomial probability formula P x C p q ( ) ( ) n x x n x = − where p is the probability of success on a single trial and q is the probability of failure on a single trial. In the formula, p will be a number between 0 and 1, inclusive, and q p 1 . = − Therefore, if p 0.2, = then q 1 0.2 0.8. = − = If p , 3 5 = then q 1 . 3 5 2 5 = − = Note that p q 1 + = and the values of p and q remain the same for each independent trial. The combination Cn x is called the binomial coefficient.
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